Triple sum frequency coherent multidimensional imaging

ABSTRACT

Methods of obtaining a multidimensional image of a sample are provided comprising (a) directing a first coherent light pulse having a first frequency ω 1  and a first wave vector k 1  at a first location in the sample, (b) directing a second coherent light pulse having a second frequency ω 2  and a second wave vector k 2  at the first location, (c) directing a third coherent light pulse having a third frequency ω 3  and a third wave vector k 3  at the first location and (d) detecting a coherent output signal having a fourth frequency ω 4  and a fourth wave vector k 4 . At least two, but optionally all three, of the coherent light pulses each excite a different transition to a discrete quantum state (e.g., transitions to vibrational states or to electronic states) of a molecule or molecular functionality in the sample. Steps (a)-(d) are repeated at a sufficient number of other locations in the sample to provide the multidimensional image.

CROSS-REFERENCE TO RELATED APPLICATIONS

The present application claims priority to U.S. Provisional PatentApplication No. 61/885,069 that was filed Oct. 1, 2013, the entirecontent of which is hereby incorporated by reference.

REFERENCE TO GOVERNMENT RIGHTS

This invention was made with government support under 1057896 awarded bythe National Science Foundation. The government has certain rights inthe invention.

BACKGROUND

The study of molecular properties in complex chemical and biologicalsamples requires sufficient spatial resolution as well as chemicalselectivity. To achieve this, certain spectroscopic techniques have beencombined with microscopy in order to investigate the properties ofmolecules in chemical and biological samples, including live cells. Forexample, chemically selective three-dimensional imaging of samples hasbeen accomplished using two-photon fluorescence microscopy. Intwo-photon fluorescence microscopy, fluorophore labeled samples areimaged on a laser scanning microscope by a tightly focused femtosecondpulsed laser having a wavelength in the near-infrared region. However,photobleaching and chemical perturbations inherent in fluorophorelabeling are limitations of the technique.

As another example, chemically selective three-dimensional imaging ofsamples has been accomplished used multiphoton vibrational microscopybased on coherent anti-Stokes Raman scattering (CARS). In CARS, a pumpfield, a Stokes field and a probe field interact with a sample togenerate an anti-Stokes field at the frequency ω_(as)=2ω_(p)−ω_(s) andwave vector k_(as)=2k_(p)−k_(s). In CARS microscopy, samples are imagedon a laser scanning microscope by a tightly focused pump laser pulse andStokes laser pulse. (The pump field and probe field are generallyderived from the same laser pulse.) Although CARS microscopy avoids thephotobleaching problem and is a label-free method, the drawbacks of thetechnique include an intrinsically weak anti-Stokes signal andsignificant nonresonant background interference, both of which limitimage contrast and spectral selectivity.

SUMMARY

Provided are methods for obtaining multidimensional images of samplesand apparatuses for carrying out the methods.

In one aspect, a method of obtaining a multidimensional image of asample is provided comprising directing a first coherent light pulsehaving a first frequency ω₁ and a first wave vector k₁ at a firstlocation in a sample, directing a second coherent light pulse having asecond frequency ω₂ and a second wave vector k₂ at the first location,directing a third coherent light pulse having a third frequency ω₃ and athird wave vector k₃ at the first location and detecting a coherentoutput signal having a fourth frequency ω₄ and a fourth wave vector k₄from the first location, wherein ω₄=±ω₁±ω₂±ω₃ and k₄=±k₁±k₂±k₃, whereinat least two of the coherent light pulses each are configured to excitea different transition to a discrete quantum state of a molecule ormolecular functionality in the sample and further wherein steps (a)-(d)are repeated at a sufficient number of other locations in the sample toprovide the multidimensional image.

In another aspect, a scanning microscope for obtaining amultidimensional image of a sample is provided optics configured toreceive coherent light pulses and to direct the coherent light pulses toa first location in the sample. The coherent light pulses comprise afirst coherent light pulse having a first frequency ω₁ and a first wavevector k₁, a second coherent light pulse having a second frequency ω₂and a second wave vector k₂, and a third coherent light pulse having athird frequency ω₃ and a third wave vector k₃, wherein at least two ofthe coherent light pulses each are configured to excite a differenttransition to a discrete quantum state of a molecule or molecularfunctionality in the sample. The scanning microscope further comprises astage configured to support the sample; and a detector positioned todetect a coherent output signal generated from the first location, thecoherent output signal having a fourth frequency ω₄ and a fourth wavevector k₄, wherein ω₄=±ω₁±ω₂±ω₃ and k₄=±k₁±k₂±k₃. The scanningmicroscope is further configured to illuminate a sufficient number ofother locations in the sample with the three coherent light pulses toprovide the multidimensional image.

Other principal features of the disclosed subject matter will becomeapparent to those skilled in the art upon review of the followingdrawings, the detailed description, and the appended claims.

BRIEF DESCRIPTION OF THE DRAWINGS

Illustrative embodiments of the disclosed subject matter will hereafterbe described with reference to the accompanying drawings.

FIG. 1A depicts the wave-mixing energy level (WMEL) diagrams for TripleSum Frequency (TSF). FIG. 1B depicts a WMEL diagram for double quantumfour-wave mixing (FWM). FIG. 1C depicts a WMEL diagram for doublequantum four-wave mixing (FWM). FIG. 1D depicts a WMEL diagram forDoubly Vibrationally Enhanced (DOVE)-Raman. FIG. 1E depicts a WMELdiagram for DOVE-IR. FIG. 1F depicts a WMEL diagram for DOVE-IR. Thenumbers label the excitation pulse frequencies and have no correlationwith their time ordering.

FIG. 2A shows benzene's high-resolution infrared absorption spectrum(provided by Bertie). FIG. 2B shows benzene's Raman spectrum, with theinset expanded around the 2ν₁₃ overtone region (all assignments followHerzberg notation). (See, Bertie, J. E. John Bertie's Download Site.2011; http://www.ualberta.ca/jbertie/JBDownload.HTM.) Data are in black,individual Lorentzian states are in blue, and fitted sums are in red.

FIG. 3 shows a 2D TSF scan of benzene (intensity level) at τ₂₁=1.2 ps,τ31=1 ps. The WMEL diagram shows the time ordering of the threeexcitation pulse resonances from left to right. The numbers identify theexcitation frequency of each pulse and the pairs of numbers identify thetime delay between the pulses. Raman and IR spectra of FIG. 2 arealigned with the ω₁ and ω₂ axes, with the Raman spectrum shifteddownward by the ν₁₃ fundamental frequency, 1478 cm⁻¹ (ω₂ ⁰). Benzenenormal mode images are taken from Page et al., and illustrate theungerade nature of the benzene modes along ω₂ and the gerade nature ofthe modes accessed by ω₁+ω₂. (See, Page, R. H.; Shen, Y. R.; Lee, Y. T.Infrared-Ultraviolet Double-Resonance Studies of Benzene Molecules in aSupersonic Beam. J. Chem. Phys. 1988, 88, 5362-5376.)

FIG. 4A is a schematic that describes pathways observed along each axisin a 2D delay scan. In observation of only {right arrow over(k)}₁+{right arrow over (k)}₂+{right arrow over (k)}₃ phase-matchingoutput, the red and/or blue arrows trace out the fundamental vibrationaldephasing rate depending on whether the first interaction involves ω₂ orω₁, respectively, and the green arrow traces out the overtone or coupledstate dephasing rate.

FIG. 4B shows a delay scan of a 15 nm benzene sample with ω₁=ω₂=1475cm⁻¹ where 2{right arrow over (k)}₁+{right arrow over (k)}₃, 2{rightarrow over (k)}₂+{right arrow over (k)}₃, and {right arrow over(k)}₁+{right arrow over (k)}₂+{right arrow over (k)}₃ TSF signals areall collected. The vertical stripe represents 2{right arrow over(k)}₁+{right arrow over (k)}₃ as it occurs at τ₃₁=0 and the diagonalstripe represents 2{right arrow over (k)}₂+{right arrow over (k)}₃ as ittraces out τ₂₁=τ₃₁.

FIG. 5 shows a 2D TSF scan collected at τ₂₁=1.5 ps, τ₃₁=3 ps, presentedas (Int)^(1/2) for clarity. Both {right arrow over (k)}₁+{right arrowover (k)}₂+{right arrow over (k)}₃ and 2{right arrow over (k)}₂+{rightarrow over (k)}₃ TSF signal has been admitted to the detector (WMELdiagrams to right). Black dots indicate Gaussian peak maxima of each ω₁slice; these are expanded in the inset and fit to another Gaussian. Thecross-peak maximum is corrected further by subtraction of the 2ω₂+ω₃line.

FIG. 6A shows a Wigner plot collected with τ₂₁=1 ps, ω₁=1480 cm⁻¹, andω_(m)=ω₁+ω₂+ω₃. FIG. 6B shows a model of the same, using dephasing ratespredicted in Table 1.

FIG. 7A shows the line shape of TSF overtone peak based solely uponα₁*α₂ enhancements of Equation 5 without M-factor. Fundamental lineshape α₁ taken from Bertie, and excited state absorption line shape (α₂)drawn from the Raman spectrum (FIG. 2B). (See, Bertie, J. E. JohnBertie's Download Site. 2011;http://www.ualberta.ca/jbertie/JBDownload.HTM.)

FIG. 7B shows the line shape of the TSF overtone peak incorporatingM-factor at 15 μm, as well as enhancements, i.e. (Equation 6)*α₁α₂l² asa representation of Equation 7. Index of refraction values provided byBertie. (See, Bertie, J. E. John Bertie's Download Site. 2011;http://www.ualberta.ca/jbertie/JBDownload.HTM; Bertie, J. E.; Jones, R.N.; Keefe, C. D. Infrared Intensities of Liquids 0.12. Accurate OpticalConstants and Molar Absorption Coefficients Between 6225 and 500 cm−1 ofBenzene at 25° C., from Spectra Recorded in Several Laboratories. Appl.Spectrosc. 1993, 47, 891-911 and Bertie, J. E.; Lan, Z. The RefractiveIndex of Colorless Liquids in the Visible and Infrared—Contributionsfrom the Absorption of Infrared and Ultraviolet Radiation and theElectronic Molar Polarizability Below 20500 cm⁻¹ . J. Chem. Phys. 1995,103, 10152-10161.)

FIG. 8A shows TSF slices along ω₁ were collected with ω₂=1480 cm⁻¹,ω_(n), =ω₁+ω₂+ω₃, and τ₂₁=τ₃₁=0 ps. FIG. 8B shows a model of intensityas a function of path length inclusive of enhancement and M-factor. Theresult was convoluted along ω₁ with a 20 cm⁻¹ Gaussian pulse to mimicexperimental data.

FIG. 9 depicts a schematic of an illustrative system for carrying outtriple sum frequency (TSF) or triply resonant sum frequency (TRSF)spectroscopy.

FIG. 10 depicts a schematic of a TSF spectroscopic analysis of benzeneaccording to an illustrative embodiment.

FIG. 11 shows a triply resonant sum frequency (TRSF) spectrum of 300 μMStyryl 9M with 180 mM benzene in DACN solution, 25 μm path length. τ₁₂=0ps, τ₃₂=2.5 ps.

FIG. 12A shows the TRSF spectrum of 300 μM Styryl 9M with 180 mM benzenein DACN solution, 25 μm path length. τ₂₁=0 ps, τ₃₂=2.5 ps (taken fromFIG. 11). FIG. 12B shows the TRSF spectrum of 2 mM Styryl 9M in DACNsolution, 25 μm path length. τ₂₁=1.75 ps and τ₃₂=2 ps.

FIG. 13 shows a Wigner scan of 2 mM Styryl 9M with ω₂=1510 cm⁻¹, τ₃₂=0ps, ω₃ power ˜1 mW. The inset shows the 1418-1500 cm⁻¹, −3.5 to −0.5 ps,region of Wigner scan with the same conditions except τ₃₂=2 ps, ω₃ power˜4 mW. Vertical slices at <1310 cm⁻¹ had τ₁₂ values shifted up 0-0.5 psto correct for inaccuracies in delay calibration in this region. Eachspectrum is plotted on the amplitude level, √{square root over (Int)}.

FIG. 14 depicts the wave-mixing energy level (WMEL) diagrams foradditional CMDS excitation schemes.

FIG. 15 depicts a schematic of a TRSF spectroscopic analysis of Styryl9M according to an illustrative embodiment.

FIG. 16 depicts a schematic of a scanning microscope for carrying outthe disclosed methods for obtaining multidimensional images of samples.

DETAILED DESCRIPTION

Provided are methods for obtaining multidimensional images of samplesand apparatuses for carrying out the methods.

The methods involve the use of multiple (e.g., three) coherent pulses oflight to excite multiple (e.g., two or three) quantum states ofmolecules or molecular functionalities present in a sample. Coherentexcitation of the multiple quantum states via the interaction of thecoherent light pulses with the electric dipoles of the molecules ormolecular functionalities generates a coherent output signal. Thecoherent output signal is greatly enhanced when the coherent excitationpulses are resonant with transitions between quantum states. Multiplelocations in a sample (i.e., locations defined by unique sets of x, y, zcoordinates) can be illuminated with spatially overlapped coherentexcitation pulses and the coherent output signal detected at eachlocation to provide a multidimensional (e.g., 2D or 3D) image of thesample. Multiresonant excitation both enhances the intensity of thecoherent output signal and suppresses background (since quantum statetransitions are specific to the different molecules or molecularfunctionalities in the sample). Thus, multidimensional images obtainedusing the disclosed methods may exhibit improved contrast as compared toimages obtained with conventional imaging methods even when theconcentration of the excited molecules or molecular functionalities inthe sample is very low.

The methods may be used to provide chemically selective,multidimensional images of a variety of types of chemical and biologicalsamples. Such samples will include different molecules orfunctionalities on molecules (i.e., molecular functionalities, e.g., C—Hbonds) having certain vibrational and electronic characteristics. Thesemolecules or molecular functionalities can be identified and studied byusing a set of coherent excitation pulses to probe these vibrational andelectronic characteristics. The molecules or molecular functionalitiesreferred to herein may be “target” molecules or molecularfunctionalities in that their existence in a particular sample may beinitially unknown. The methods can be used to determine the presence orabsence of such molecules/molecular functionalities in the samples. Themethods will find applications in a variety of fields including membranebiology, neurobiology, pathology, pharmacology and composite materials.

The disclosed methods employ various coherent multidimensionalspectroscopic (CMDS) excitation schemes. The field of coherentmultidimensional spectroscopy (CMDS) uses a series of excitation pulsesthat are bright enough to create time-dependent multiple quantumcoherences (MQCs). The MQCs launch new, directional light fields at thefrequency differences between the MQC states for the duration of thecoherence. The emission occurs only between pairs of states that arecoupled. The intensity of the emission informs the observer of couplingbetween the modes of the system and therefore describes its structure.Dynamics are explored by observing dephasing rates and changes incouplings over time. Because the wave vector for the nonlinear outputpolarization is the sum of the input wave vectors weighted by thepositive or negative phase of interaction, the phase of each fieldinteraction can be defined by observing the output launched in aspecific direction. That direction is described by conservation ofmomentum between the input and output fields. This phase-matchingcondition helps to define the quantum pathway represented by lightemitted in a certain direction.

Vibrational CMDS is divided into two different approaches.Two-dimensional infrared or electronic spectroscopy (2D-IR or 2D-ES) aretime domain Four Wave Mixing (FWM) methods where the wide bandwidth ofultrafast excitation pulses excites multiple quantum states and thedelay times between pulses map out the frequencies and decay times ofthe coherences through their temporal interference. Multiresonant CMDSis a frequency domain method where the excitation pulses are long enoughto excite individual quantum states but short enough to map out theirdynamics. Both methods are capable of using fully coherent pathwayswhere populations are not created. Multiresonant methods access a widerrange of quantum states because they only require short-term phasecoherence during the nonlinear mixing. This difference removes theconstraint imposed by the excitation bandwidth and allows any set ofvibrational and electronic states to be probed, regardless of frequency.

In the time domain methods (e.g., 2D-IR and 2D-ES), the experimentsgenerally employ pulses of duration of ˜50 fs and bandwidth ˜300 cm⁻¹.The phase oscillations of each coherence are resolved by heterodyningthe output with a local oscillator, scanning the time delays betweenpulses, and Fourier transforming the signals to the frequency domain. Inmultiresonant CMDS experiments, pulses are designed to have spectral andtemporal width matching the dephasing times of interest in the sample(often ˜4 ps, ˜20 cm⁻¹ for vibrational spectroscopy). With this design,each pulse has the spectral width to excite single modes, and temporallydefine the sequence of interactions between quantum states. Frequenciesare scanned to obtain spectral information and pulses are delayed to mapout dynamics. Multiresonant CMDS can use any combination of electronicand vibrational states.

In one embodiment, a method of obtaining a multidimensional image of asample comprises (a) directing a first coherent light pulse having afirst frequency ω₁ and a first wave vector k₁ at a first location in asample, (b) directing a second coherent light pulse having a secondfrequency ω₂ and a second wave vector k₂ at the first location, (c)directing a third coherent light pulse having a third frequency ω₃ and athird wave vector k₃ at the first location and (d) detecting a coherentoutput signal having a fourth frequency ω₄ and a fourth wave vector k₄from the first location. In the method, at least two of the coherentlight pulses each are configured to excite a different transition to adiscrete quantum state of a molecule or molecular functionality in thesample. The term “discrete” is used to distinguish the quantum statefrom a virtual quantum state. In the method, steps (a)-(d) are repeatedat a sufficient number of other locations in the sample to provide themultidimensional image.

The transitions excited by at least two of the coherent light pulses maybe transitions to vibrational quantum states (e.g., a fundamental mode,an overtone, or a combination band) or to electronic quantum states. Oneof the two coherent light pulses may excite a transition to avibrational quantum state and the other may excite a transition to adifferent vibrational quantum state. Alternatively, one of the twocoherent light pulses may excite a transition to a vibrational quantumstate and the other may excite a transition to an electronic quantumstate. The excitations achieved by the two coherent light pulses may besingle-quantum excitations or multiple-quantum, e.g., two-quantumexcitations.

As discussed above, at least two of the coherent light pulses eachexcite a different transition to a discrete quantum state of a moleculeor molecular functionality in the sample. The remaining coherent lightpulse may be configured to excite a transition to a virtual quantumstate (e.g., a virtual electronic state), whereby a Raman transition isinduced returning the molecule or molecular functionality to a lowerenergy state (e.g., the ground state). Alternatively, the remainingcoherent light pulse may excite a transition to a discrete quantum state(e.g., an electronic quantum state), whereby a “resonance Raman”transition is induced returning the molecule or molecular functionalityto a lower energy state. Either Raman transition may be a single-quantumor a multiple-quantum transition. The excitation may be a single-photonexcitation.

Examples of specific CMDS excitation schemes are shown in FIG. 1 anddescribed in the Examples below. FIG. 14 includes other examples ofexcitation schemes. Pathways II and III are DOVE-IR and DOVE-Ramanpathways (see FIG. 1D-F). Pathway VI is a triply resonant sum frequency(TRSF) pathway. Pathways I, IV, and VII are similar to DOVE-IR,DOVE-Raman, and TRSF, respectively, but substitute a second electronictransition for the combination band transition in the infrared, givingdifferent selectivity. Pathway V is a fully infrared pathway.

The coherent light pulses may be characterized by a number ofproperties, such as the frequency, ω, of each light pulse. The selectedfrequency generally depends upon the desired CMDS excitation scheme aswell as the molecule or molecular functionality to be imaged. Thefrequency of a coherent light pulse may be fixed or may be tuned (i.e.,scanned) over a range of frequencies. Again, the selected rangegenerally depends upon the desired CMDS excitation scheme as well as themolecule or molecular functionality to be imaged. In order to enhancethe coherent output signal, the selected frequency can be that which isresonant with the transition to be excited with the coherent light pulse(or the range of frequencies is selected to encompass the resonantfrequency). The frequency of a coherent light pulse may be independentlytunable from another coherent light pulse, i.e., the selection of arange of frequencies over which one coherent light pulse is to be tuneddoes not depend upon the selected frequency of another coherent lightpulse or the range of frequencies over which the other coherent lightpulse is tuned. Suitable frequencies and ranges of frequencies includethose in the infrared region (e.g., from about 10⁴ cm⁻¹ to about 10cm⁻¹), those in the visible region (e.g., from about 2.5×10⁴ cm⁻¹ toabout 10⁴ cm⁻¹), and those in the ultraviolet region (e.g., from about10⁶ cm⁻¹ to 2.5×10⁴ cm⁻¹) of the electromagnetic spectrum. For multiplecoherent light pulses, e.g., a first coherent light pulse having a firstfrequency ω₁, a second coherent light pulse having a second frequencyω₂, and a third coherent light pulse having a third frequency ω₃, thesubscripts are meant only to distinguish individual, independentcoherent light pulses from one another. The selected frequency or rangeof frequencies for one coherent light pulse (e.g., a first coherentlight pulse having a first frequency ω₁) may be the same or different asanother coherent light pulse (e.g., a second coherent light pulse havinga second frequency ω₂).

The coherent light pulses may be characterized by a number of otherproperties including spectral width and temporal width, which aregenerally selected based upon the desired CMDS excitation scheme. Thecoherent light pulses may be characterized by pulse energy, which isgenerally selected to provide sufficient intensity to ensure nonlinearinteractions with certain molecules or molecular functionalities withinthe sample. The coherent light pulses may be characterized by repetitionrate, which is generally selected to provide a desired acquisition speedfor the image. Another property characterizing the coherent light pulseis its wave vector, k.

A coherent light pulse may be characterized by the orientation of itspropagation axis relative to normal to the plane of the sample. Forexample, a coherent light pulse may have a propagation axis which isnormal to the sample plane. Alternatively, a coherent light pulse mayhave a propagation axis which forms an angle θ relative to normal. Thecoherent light pulses may be configured in a non-collinear beam geometryin which different coherent light pulses are characterized by differentpropagation axes. (See, e.g., FIGS. 9 and 10.)

The coherent light pulses may be characterized by the time delaysbetween different coherent light pulses. By way of example only, timedelays may be defined relative to a first coherent light pulse such thatτ₂₁=τ₂−τ₁ and τ₃₁=τ₃−τ₁. (See, e.g., Example 1.) The time delaysdetermine the order in which the coherent light pulses interact with thesample and in general, any order may be used. Coherent light pulses maybe considered to be temporally overlapped when the time delay betweenthe coherent light pulses is substantially zero. The time delay betweencoherent light pulses may be fixed (e.g., to collect a frequencyspectrum with a certain pathway) or may be scanned over a range of timedelays (e.g., to learn dynamics).

Various embodiments of the method may be conducted in which thefrequencies of the coherent light pulses are fixed or tuned and in whichthe time delays between different coherent light pulses are fixed orscanned. By way of example only, in some embodiments, the frequencies ofeach of the coherent light pulses are fixed and the time delays betweendifferent coherent light pulses are fixed. Such an embodiment may beuseful for obtaining a “chemical map” of a specific known molecule ormolecular functionality in a sample. The Examples, below, describe otherembodiments of the method.

As discussed above, the coherent light pulses are spatially overlappedat various locations on or within the sample. The coherent light pulsesmay be configured such that the pulses are substantially fullyoverlapped in space, i.e., the centers of each spot illuminated by itsrespective coherent light pulse are substantially coincident. However,the coherent light pulses may be configured such that the pulses arepartially spatially overlapped, i.e., the centers of one or more spotsilluminated by its respective coherent light pulse are not coincident. Apartially spatially overlapped configuration may allow for the spatialregion probed by the coherent light pulses to be smaller than thesmallest diffraction limited spot size.

As discussed above, the multiple coherent light pulses interact withmolecules or molecular functionalities in a sample to generate anonlinear output polarization, which acts as the source of radiation fora coherent output signal having a specific frequency and wave vector asdetermined by the CMDS excitation scheme, the phase of interaction andconservation of momentum. In addition, for a particular CMDS excitationscheme, different coherent output signals having different frequenciesand wave vectors associated with different quantum pathways may bepossible. Particular quantum pathway(s) may be monitored by detectingthe coherent output signal(s) in the corresponding phase-matcheddirection(s). The detection of the desired coherent output signal anddiscrimination from other possible coherent output signals may befacilitated by placing the detector coincident with the desiredphase-matched direction, by using certain beam geometries, and by usingapertures to physically block undesired signals. In addition, alteringthe propagation axes of the coherent light pulses and therefore, thepropagation axis of the corresponding coherent output signal allows fordifferent pathways, such as those in FIGS. 1 and 14, to bephase-matched. Interrogating different pathways can provide differentinformation about the chemical makeup of the sample.

In some embodiments, a method of obtaining a multidimensional image of asample comprises (a) directing a first coherent light pulse having afirst frequency ω₁ and a first wave vector k₁ at a first location in asample, (b) directing a second coherent light pulse having a secondfrequency ω₂ and a second wave vector k₂ at the first location, (c)directing a third coherent light pulse having a third frequency ω₃ and athird wave vector k₃ at the first location and (d) detecting a coherentoutput signal having a fourth frequency ω₄ and a fourth wave vector k₄from the first location, wherein ω₄=±ω₁±ω₂±ω₃ and k₄=±k₁±k₂±k₃. In themethod, two of the coherent light pulses are each configured to excite adifferent transition to a discrete quantum state (e.g., differentvibrational quantum states or a vibrational quantum state and anelectronic quantum state) of a molecule or molecular functionality inthe sample. The remaining coherent light pulse may be configured toexcite a transition to a virtual quantum state (e.g., a virtualelectronic state), whereby a Raman transition is induced returning themolecule or molecular functionality to a lower energy state (e.g., theground state). Alternatively, the remaining coherent light pulse may beconfigured to excite a transition to a discrete quantum state (e.g., anelectronic quantum state), whereby a “resonance Raman” transition isinduced to return the molecule or molecular functionality to a lowerenergy state. In the method, steps (a)-(d) are repeated at a sufficientnumber of other locations in the sample to provide the multidimensionalimage.

In some embodiments, a method of obtaining a multidimensional image of asample comprises (a) directing a first coherent light pulse having afirst frequency ω₁ and a first wave vector k₁ at a first location in asample, (b) directing a second coherent light pulse having a secondfrequency ω₂ and a second wave vector k₂ at the first location, (c)directing a third coherent light pulse having a third frequency ω₃ and athird wave vector k₃ at the first location and (d) detecting a coherentoutput signal having a fourth frequency ω₄ and a fourth wave vector k₄from the first location, wherein ω₄=ω₁+ω₂+ω₃ and k₄=k₁+k₂+k₃. In themethod, steps (a)-(d) are repeated at a sufficient number of otherlocations in the sample to provide the multidimensional image. Asdiscussed above, two of the coherent light pulses may each be configuredto excite a different transition to a discrete quantum state (e.g.,different vibrational quantum states or a vibrational quantum state andan electronic quantum state) of a molecule or molecular functionality inthe sample. The remaining coherent light pulse may be configured toexcite a transition to a virtual quantum state (e.g., a virtualelectronic state), whereby a Raman transition is induced returning themolecule or molecular functionality to a lower energy state (e.g., theground state). Alternatively, the remaining coherent light pulse may beconfigured to excite a transition to a discrete quantum state (e.g., anelectronic quantum state), whereby a “resonance Raman” transition isinduced to return the molecule or molecular functionality to a lowerenergy state. These embodiments may be referred to as triple sumfrequency (TSF) schemes (if they involve excitation of two discretequantum states) or triply resonant sum frequency (TRSF) schemes (if theyinvolve excitation of three discrete quantum states).

In some embodiments, one of the coherent light pulses excites atransition to a vibrational quantum state (e.g., a fundamental mode),one of the coherent light pulses excites a transition to a differentvibrational quantum state (e.g., an overtone or combination band), andone of the coherent light pulses excites a transition to a virtualelectronic state, whereby a Raman transition is induced returning themolecule or molecular functionality to a lower energy state. In someembodiments, one of the coherent light pulses excites a transition to avibrational quantum state (e.g., a fundamental mode), one of thecoherent light pulses excites a transition to a different vibrationalquantum state (e.g., an overtone or combination band), and one of thecoherent light pulses excites a transition to an electronic quantumstate, whereby a “resonance Raman” transition is induced returning themolecule or molecular functionality to a lower energy state. In someembodiments, one of the coherent light pulses excites a transition to avibrational quantum state, one of the coherent light pulses excites atransition to a electronic quantum state, and one of the coherent lightpulses excites a transition to a different electronic quantum state,whereby a “resonance Raman” transition is induced returning the moleculeor molecular functionality to a lower energy state.

The Examples below further describe methods employing TSF and TRSFexcitation schemes.

As discussed above, methods employing TSF or TRSF excitation schemesinvolve detecting a coherent output signal having a fourth frequency ω₄and a fourth wave vector k₄, wherein ω₄=ω₁+ω₂+ω₃ and k₄=k₁+k₂+k₃.Because this quantum pathway cannot be phase matched in systems withnormal dispersion, the inventors' findings that the correspondingcoherent output signal was not only detectable but also more intensethan coherent output signals obtained using other CMDS excitationschemes are unexpected. The inability to achieve phase matching for thispathway means that there is no beam geometry that can prevent thecoherent output signal from molecules near the front of the sample frominterfering destructively with the coherent output signal from moleculesat the opposite end of the sample. However, the inventors found that thegain of this pathway (i.e., the ability for the pathway to generatesignal as a function of path length, the distance over which thecoherent light pulses interact with molecules or molecularfunctionalities in the sample) was much higher than expected. Moreover,the inventors have found that the path length is an important parameterthat affects the intensity of the coherent output signal in thek₄=k₁+k₂+k₃ direction. Generally, the path length should be sufficientlylarge to maximize the gain, but sufficiently small to minimize theproblem of the destructive interference of the coherent output signal.The range of suitable path lengths depends upon a number of otherparameters, including the particular sample and the particular quantumstate transitions to be excited. However, in some embodiments, the pathlength is in the range of from about 1 μm to about 200 μm. This includesembodiments in which the path length is in the range of from about 1 μmto about 150 μm, from about 1 μm to about 100 μm or from about 1 μm toabout 50 μm.

The methods may be carried out on a variety of scanning microscopes,including confocal scanning microscopes. In one embodiment, the scanningmicroscope comprises optics configured to receive coherent light pulsesfrom light sources and to direct the pulses into a sample. These opticsmay include optics configured to direct the coherent light pulses alongdesired propagation axes to achieve a desired beam geometry (i.e., acollinear beam geometry or a non-collinear beam geometry) and optics(e.g., an objective lens) configured to focus the coherent light pulsesinto the sample. The scanning microscope also comprises a stageconfigured to support the sample and a detector (e.g., a photomultipliertube) configured to receive and to detect a coherent output signalgenerated from the sample.

The scanning microscope may comprise a variety of other components knownto be useful for scanning microscopy. The scanning microscope maycomprise the light sources (and associated optics) configured togenerate coherent light pulses having certain of the characteristicsdescribed herein (e.g., frequency, spectral width, temporal width, pulseenergy). The scanning microscope may comprise optics configured toadjust the time delay between the coherent light pulses. The scanningmicroscope may comprise optics configured to receive light generatedfrom the sample or passing through the sample and to direct the lighttowards a detector, including optics configured to focus or collimatethe light. The scanning microscope may comprise an aperture configuredto receive light generated from the sample or passing through the sampleand to block undesired light (e.g., certain coherent excitation pulsesor undesired coherent output signals). This or another aperture may alsobe configured to block light generated from regions within the samplewhich are outside the focus region, although such apertures may not benecessary due to the improvement in contrast obtained via the use ofmultiresonant excitation. The scanning microscope may comprise opticalfilters configured to receive light generated from the sample or passingthrough the sample and to block undesired light (e.g., certain coherentexcitation pulses). Monochromators may also be used for this purpose.

Obtaining a multidimensional image of the sample using the scanningmicroscope requires illuminating multiple locations in a sample withspatially overlapped coherent excitation pulses and detecting thecoherent output signal generated at each location. This may be achievedby scanning the coherent excitation pulses relative to the sample, e.g.,via scanning optics in the scanning microscope, or by scanning thesample relative to the coherent excitation pulses, e.g., via a scanningstage. In the former case, the scanning optics typically control the xand y position of the coherent excitation pulses relative to the sample.The z position may be controlled by adjusting the position of theobjective lens relative to the sample. The scanning optics for scanningthe coherent excitation pulses relative to the sample may includegalvano scanners, e.g., such as those available on the Nikon A1 MPconfocal microscope available from Nikon Instruments, Inc. The spatialresolution that is possible in the scanning microscope is determined bythe diffraction limited spot size of the highest frequency of thecoherent excitation pulses and the coherent output signal.

FIG. 16 shows an exemplary embodiment of a scanning microscope 1600 forcarrying out any of the disclosed methods. The scanning microscope 1600may include a coherent multidimensional spectroscopy (CMDS) laser system1602 which may include the light sources and associated opticsconfigured to generate coherent light pulses 1604 a-c having certain ofthe characteristics described herein. The scanning microscope 1600 mayinclude a dichroic mirror 1606 configured to direct the coherent lightpulses towards a sample on a stage 1608 configured to support thesample. The scanning microscope 1600 may include a laser scanning system1610 which may include scanning optics for scanning the coherentexcitation pulses 1604 a-c relative to the sample. The scanningmicroscope 1600 may include a microscope objective 1612 configured tofocus the coherent light pulses 1604 a-c into the sample. The scanningmicroscope 1600 may include a pinhole aperture 1614 configured toreceive light generated from the sample or passing through the sampleand to block undesired light from impinging upon a photomultiplier tube1616 configured to receive and to detect a coherent output signal 1618generated from the sample.

The scanning microscope may further comprise components for controllingcertain operations of the scanning microscope, e.g., the illumination ofdifferent locations within the sample with the coherent excitationpulses or the reconstruction of a multidimensional image from thecoherent output signal detected at the different locations. For example,the scanning microscope may further comprise a processor and acomputer-readable medium operably coupled to the processor, thecomputer-readable medium having computer-readable instructions storedthereon that, when executed by the processor cause the scanningmicroscope to perform certain operations, e.g., operations related toilluminating a particular location within the sample or toreconstructing an image from detected signals. For example, the scanningmicroscope may further comprise a processor and a non-transitorycomputer-readable medium operably coupled to the processor, thecomputer-readable medium having computer-readable instructions storedthereon that, when executed by the processor cause the scanningmicroscope to illuminate a first location within the sample with a firstcoherent light pulse having a first frequency ω₁ and a first wave vectork₁, a second coherent light pulse having a second frequency ω₂ and asecond wave vector k₂, and a third coherent light pulse having a thirdfrequency ω₃ and a third wave vector k₃ to generate a first coherentoutput signal characterized by ω₄=±ω₁±ω₂±ω₃ and k₄=±k₁±k₂±k₃; receivethe first coherent output signal from the detector; illuminate multipleother locations within the sample with the three coherent light pulsesto generate multiple other coherent output signals, each characterizedby ω₄=±ω₁±ω₂±ω₃ and k₄=±k₁±k₂±k₃; reconstruct a multidimensional imagefrom the received coherent output signals; and output themultidimensional image.

The methods will be understood more readily by reference to thefollowing examples, which are provided by way of illustration and arenot intended to be limiting.

EXAMPLES Example 1 Fully Coherent Triple Sum Frequency Spectroscopy of aBenzene Fermi Resonance

This example is derived from Boyle et al. Fully Coherent Triple SumFrequency Spectroscopy of a Benzene Fermi Resonance, J. Phys. Chem. A2013, 117, 5578-5588 and accompanying supporting information, each whichis hereby incorporated by reference in its entirety.

Introduction

The two dominant forms of multiresonant vibrational CMDS are DoublyVibrationally Enhanced (DOVE) CMDS and Triply Vibrationally Enhanced(TRIVE) CMDS. Each approach uses three excitation pulses and the phasematching condition {right arrow over (k)}₁−{right arrow over(k)}₂+{right arrow over (k)}₃, where the subscripts designate thefrequencies. DOVE CMDS uses two infrared pulses to excite vibrationalstates and a higher frequency third pulse to excite a Raman transitionbetween the vibrational states. One of the infrared transitions is atwo-quantum transition directly to an overtone or combination band. Theoutput at ω₁−ω₂+ω₃ appears in the visible or ultraviolet and isspectrally distinct from the excitation frequencies. TRIVE CMDS usesthree infrared pulses to excite vibrational states and the outputappears in the infrared. The Klug group has applied DOVE to benzene in acomparable spectral region to that presented in this example, offeringan interesting comparison of the two methods. (See, Donaldson, P. M.;Guo, R.; Fournier, F.; Gardner, E. M.; Barter, L. M. C.; Barnett, C. J.;Gould, I. R.; Klug, D. R.; Palmer, D. J.; Willison, K. R. DirectIdentification and Decongestion of Fermi Resonances by Control of PulseTime Ordering in Two-dimensional IR Spectroscopy. J. Chem. Phys. 2007,127, 114513.) For a centrosymmetric molecule such as benzene, it turnsout that the overtone and combination bands states seen in the twomethods are mutually exclusive.

This example reports multiresonant Triple-Sum Frequency (TSF) CMDS withthe use of benzene as a model system. It uses the fully additive phasematching pathway {right arrow over (k)}₁+{right arrow over (k)}₂+{rightarrow over (k)}₃. This pathway is not able to achieve the phase matchedcondition in systems with normal dispersion, and yet the output signalis even brighter than the comparable DOVE CMDS of benzene. FIG. 1A showsTSF CMDS has only a single coherence pathway involving a simple ladderclimbing of states. It is a fully coherent parametric pathway that doesnot involve any intermediate populations and it returns the system tothe ground state. No energy is deposited in the molecular system. UnlikeDOVE, the first two transitions are each single-quantum interactionsthat evolve the system first to fundamental and then to overtone orcombination band vibrational coherences. The final excitation creates aRaman transition. The output is spectrally isolated at the sum frequencyof the inputs where high-efficiency detectors are available.

Because this ladder-climbing has only a single pathway, spectralinterpretation is clear and there cannot be any interference effectsbetween pathways. The fundamental transitions appear along one axis andthe overtones and combination bands appear along the other. Thisseparation is similar to double quantum FWM, a fully vibrationaltechnique where anharmonicity can be deduced directly from the frequencyof the double quantum axis. However, double quantum FWM includesinterference between two pathways with opposite phases that involve thefundamental and an overtone or combination band (FIG. 1B,C). Thesepathways destructively interfere in the absence of coupling between thestates and lead to the disappearance of the peak. The other FWM pathwaysfor 2D-IR and TRIVE also exhibit these same interference effects betweenparametric pathways that return the system to the ground state andnonparametric pathways that involve overtones or combination bands andleave the system in an excited state. These interference effects areabsent in TSF CMDS where there is a single pathway and all peaks sharepositive phase. Enhancement in this case can be due to electrical aswell as mechanical anharmonicity. Finally, in TSF CMDS there can be nogeneration of a population state and the pathway is therefore fullycoherent. This technique therefore offers an interesting probe ofcoherence transfer because this is the only process that would create anew output frequency.

Experimental

Data were collected on a modified version of a previously describedlaser system. (See, Block, S. B.; Yurs, L. A.; Pakoulev, A. V.;Selinsky, R. S.; Jin, S.; Wright, J. C. Multiresonant MultidimensionalSpectroscopy of Surface-Trapped Excitons in PbSe Quantum Dots. J. Phys.Chem. Lett. 2012, 3, 2707-2712.) Briefly, an 80 MHz mode-lockedTi:Sapphire SpectraPhysics Tsunami femtosecond oscillator seeded aregenerative amplifier, pumped by a 1 kHz, 9 W Nd:YLF Empower. The 800nm, 1.0-1.6 ps, 12 cm⁻¹ amplifier output was split to pump twoindependently tunable optical parametric amplifiers (OPAs) withfrequencies ω₁ and ω₂ in the mid-IR, 20-25 cm⁻¹ fwhm Gaussian profiles,˜0.8-1 ps durations, and 1-1.5 μJ pulse energies. A third excitationbeam (ω₃) with a 2 μJ/pulse of 800 nm light was created by furthersplitting the amplifier output. The ω₂ and ω₃ time delays were changedrelative to the ω₁ beam by times τ₂₁ and τ₃₁. Note that in thisnotation, numerical indices refer to the beamline and not the quantumstate or time ordering. This notation allows for two-dimensional scansto be described self-consistently, not only in frequency space but indelay space as well. In the former, a fixed τ₂₁ and τ₃₁ will define thepathway that is being probed. In the latter, frequencies are fixed andrelative time delays are varied over positive and negative values inorder to probe all possible time ordered pathways. For TSF, thesetwo-dimensional delay scans map out each of the coherence dephasingrates.

All fields were focused into the sample by either a 50 or 100 mm focallength off-axis parabolic mirror. The experimental phase-matchinggeometry placed the visible beam normal to the sample plane and theinfrared fields at ±10°. The wave vector for the nonlinear outputpolarization is the sum of the excitation pulse wave vectors, {rightarrow over (k)}₄={right arrow over (k)}₁+{right arrow over (k)}₂+{rightarrow over (k)}₃, so the desired TSF output is therefore collinear tothe visible field as the momenta of the IR fields will cancel when thefrequencies are similar. It should be noted that the TSF pathways2{right arrow over (k)}₁+{right arrow over (k)}₃ and 2{right arrow over(k)}₂+{right arrow over (k)}₃ result in strong FWM as well, and thatthese outputs occur at ˜2-3° to the visible beam and are spatiallyremoved. Spectral discrimination of the signal from the visible inputwas achieved by a combination of high optical density 600 nm long-passand 800 nm notch filters. A McPherson Model 218A monochromator measuredthe TSF signal for experiments that resolved the output frequency with aresolution of 11 cm⁻¹. The OPAs and laser table are purged with dry air,maintaining a relative humidity of ˜2%.

The tuning of ω₁ and ω₂ over 200 cm⁻¹ changed the delay times by ˜0.3ps, largely as a result of the path-length change in the AgGaS₂ DFGcrystal. It was found that inducing either two-photon fluorescence ornonresonant TSF signal in a 2 mm ZnSe crystal provided a convenient andvery intense output for finding an approximate zero delay time. However,the temporal breadth of the ZnSe signal results in slight variations ofzero delay definition from day to day, which can slightly change therelative intensities of benzene's two peaks as well as change thetemporal offset in Wigner data.

The sample was neat benzene held between two 150 μm CaF₂ windows with aTeflon spacer of path length 15 μm. Infrared spectra of the sample wereidentical with those obtained with benzene dried by distillation oversodium.

Schematics of the experimental set-up are shown in FIGS. 9 and 10. InFIG. 9, the system 900 includes the light sources 902 which generate afirst coherent light pulse 904 having frequency ω₁, a second coherentlight pulse 906 having frequency ω₂, and a third coherent light pulse908 having frequency ω₃. Optics 910 are configured to provide timedelays between the coherent light pulses. Optics 912 are configured tofocus the coherent light pulses in the sample 915. Optics 914 areconfigured to direct light generated from the sample (including thecoherent output signal 916) or passing through the sample (including thecoherent excitation pulses 904, 906 and 908) towards a detector. Anaperture 918 blocks the coherent excitation pulses 904 and 906 and a setof optical filters 920 blocks the coherent excitation pulse 908. Thecoherent output signal 916 is resolved by the monochromator 922. In thesystem 900, the coherent excitation pulses 904, 906 and 908 enter thesample 915 at the entrance plane 924 and they exit the sample 915 at theoppositely facing exit plane 926, along with the coherent output signal916 generated in the sample 915.

In FIG. 10, a first coherent light pulse 1004 having frequency ω₁, asecond coherent light pulse 1006 having frequency ω₂, and a thirdcoherent light pulse 1008 having frequency ω₃ are focused into a sample1014 containing benzene. The coherent light pulses excite certainquantum states of benzene to generate a coherent output signal 1016having frequency ω₄.

Theory

In this section the formalism used to predict spectral intensity andline shape and to simulate multidimensional experiments is presented.Also discussed are the phase-matching restrictions imposed in TSFwave-mixing experiments in condensed media. Finally, comparisons aredrawn between pathways in TSF and DOVE or Double Quantum 2D-IR.

FIG. 1A shows the evolution of the states using the wave mixing energylevel (WMEL) diagram for TSF CMDS. The time evolution proceeds from leftto right. The letters designate the ground, vibrational, and virtualelectronic states, the numbers designate the excitation frequency, andthe solid and dotted arrows designate transitions that change the ket orbra states, respectively, of the density matrix elements. In therotating wave approximation where ρ_(mn)={tilde over (ρ)}_(mn)e^(−iω)^(mn) ^(t), the following Liouville equation relates the initial {tildeover (ρ)}_(pn) and {tilde over (ρ)}_(mp) density matrix amplitudes tothe final {tilde over (ρ)}_(mn) density matrix amplitude:

$\begin{matrix}{{\overset{.}{\overset{\sim}{\rho}}}_{mn} = {{{- \Gamma_{mn}}{\overset{\sim}{\rho}}_{mn}} + {\frac{}{2}\left( {{\frac{{\overset{\rightarrow}{\mu}}_{pm} \cdot \overset{\rightarrow}{E}}{\hslash}{\overset{\sim}{\rho}}_{pn}^{\; \omega_{mp}t}} - {\frac{{\overset{\rightarrow}{\mu}}_{pn} \cdot \overset{\rightarrow}{E}}{\hslash}{\overset{\sim}{\rho}}_{mp}^{\; \omega_{pn}t}}} \right)}}} & \left( {{Equation}\mspace{14mu} 1} \right)\end{matrix}$

where {right arrow over (E)}=Σ_(i)1/2E_(i)^(o)(t)(e^(i({right arrow over (k)}) ^(i) ^(·{right arrow over (z)})^(i) ^(ω) ^(i) ^(t))+e^(−i({right arrow over (k)}) ^(i)^(·{right arrow over (z)}) ^(i) ^(−ω) ^(i) ^(t))); and E_(i) ^(o),{right arrow over (k)}_(i), are the envelope, wave vector, and frequencyof the ith excitation pulse; and ω_(mn), μ_(mn), and Γ_(mn), are thefrequency, transition moment, and dephasing rate of the mn coherence.

The Liouville equations can be simply solved in the steady state toidentify the factors controlling the resonance enhancements. The steadystate approximates the spectral response for the relatively long 1 pspulse widths in these experiments. For example, the WMEL pathway shownin FIG. 1A contains the Liouville diagram gg

vg, which describes the evolution of the ground state population to a vgcoherence where v is a vibrational state and ω₁ is the excitationfrequency. In this case, the resonance enhancement for this transitiondepends on the minimization of Δ₁≡ω_(vg)−ω₁−iΓ_(vg). The Liouvillediagram vg

v′g describes the second interaction in FIG. 1A that creates a v′gcoherence using ω₂. The vg coherence frequency at the time of the secondinteraction will either be the driven frequency, ω₁, if the second pulsetemporally overlaps the first, or the free induction decay frequency,ω_(vg), if it is delayed. If the ω₂ excitation temporally overlaps withthe ω₁ excitation pulse, the vg coherence will have a driven componentat the ω₁ frequency and the resonance enhancement depends onminimization of Δ₂≡ω_(v′v)−ω₂−iΓ_(v′g). If the ω₂ excitation is delayedfrom the ω₁ excitation pulse, the vg coherence will have the freeinduction decay frequency and the resonance enhancement depends onminimization of Δ₂≡ω_(v′v)−ω₂−iΓ_(v′g).

The third interaction is described by the Liouville diagram v′g

eg where the e electronic state represents a virtual sum-over-states inthe case of an electronically nonresonant system like benzene, and theeg coherence is responsible for creating the output field. If the ω₃excitation temporally overlaps with the ω₂ excitation pulse, the v′gcoherence will have a driven component at ω₂ (or ω₁+ω₂ frequency if thethree excitation pulses overlap) and the resonance enhancement dependson minimization of Δ₃≡ω_(ev)−ω₂−ω₃ iΓ_(eg) (orΔ₃≡ω_(eg)−ω₁−ω₂−ω₃−iΓ_(eg)). If the ω₃ excitation does not overlap withthe ω₂ excitation pulse, the v′g coherence will have the free inductiondecay frequency and the resonance enhancement depends onΔ₃≡ω_(ev′)−ω₃−iΓ_(eg). The output coherence frequency will depend uponthe relative contributions from the driven and free induction decaycomponents and can range from ω₁+ω₂+ω₃ if the three excitation pulsesoverlap in time to ω_(v′g)+ω₃ if the third pulse is delayed. The thirdinteraction and the output coherence together correspond to a Ramantransition involving the gerade v′ states.

The transition moments for the infrared absorption transitions arerelated to the absorption coefficient by the equation

$\begin{matrix}{\alpha = {\frac{4\; \pi \; \omega \; {NF}_{1}\mu_{vg}^{2}\Gamma_{vg}}{\hslash \; c{\Delta_{vg}}^{2}n^{o}}\left( {\rho_{gg} - \rho_{vv}} \right)}} & \left( {{Equation}\mspace{14mu} 2} \right)\end{matrix}$

where N is the number of oscillators, F is the local field enhancement,ω is the infrared angular frequency, and n^(o) is the refractive index.The integrated Raman cross-section is related to the transition momentsand the resonance denominator involved in the electronic transitions bythe equation

$\begin{matrix}{\sigma = \frac{F_{4}n_{{ev}^{\prime}}\omega_{{ev}^{\prime}}^{4}\mu_{{ev}^{\prime}}^{2}\mu_{eg}^{2}\rho_{gg}d\; \Omega}{2\; \hslash^{2}c^{4}n_{eg}{\Delta_{eg}}^{2}}} & \left( {{Equation}\mspace{14mu} 3} \right)\end{matrix}$

where Ω is the solid angle of collection. The third order susceptibilityfor TSF-FWM is related to these same quantities by the equation

$\begin{matrix}{\chi^{(3)} = \frac{{NF}_{4}\mu_{eg}\mu_{gv}\mu_{{vv}^{\prime}}\mu_{v^{\prime}e}\rho_{gg}}{4\; D\; \hslash^{3}\Delta_{1}\Delta_{2}\Delta_{3}}} & \left( {{Equation}\mspace{14mu} 4} \right)\end{matrix}$

where D=6 in the Maker-Terhune convention. The TSF-FWM output intensityin the steady state limit depends on |χ⁽³⁾|² and is defined by

$\begin{matrix}{I = {\frac{256\; \pi^{4}\omega_{4}^{2}D^{2}F_{4}^{2}{\chi^{(3)}}^{2}l^{2}}{n^{o}c^{4}}{MI}_{1}I_{2}I_{3}}} & \left( {{Equation}\mspace{14mu} 5} \right)\end{matrix}$

where l is the path length, I_(i) is the intensity of the ith excitationpulse, and M is a factor that corrects for the frequency dependence ofthe absorption and phase matching effects,

$\begin{matrix}{M = \frac{{^{{- \alpha_{4}}l}\left\lbrack {1 - ^{\Delta \; \alpha \; {l/2}}} \right\rbrack}^{2} + {4\; ^{\Delta \; \alpha \; {l/2}}{\sin \left\lbrack {\Delta \; {{kl}/2}} \right\rbrack}^{2}}}{\left( {\Delta \; \alpha \; {l/2}} \right)^{2} + \left( {\Delta \; {kl}} \right)^{2}}} & \left( {{Equation}\mspace{14mu} 6} \right)\end{matrix}$

where Δα≡α₄−(α₁+α₂+α₃) and α_(i) are the absorption coefficients at theith excitation or output frequency. In the limit where absorption andrefractive index dispersion are negligible, the M factor reduces to thenormal sin c² (Δkl/2) dependence on the phase mismatch. SubstitutingEquations 2, 3, 4, and 6 into Equation 5 and fixing α₃=α₄=0, as is truein the current experiment, provides an expression that describes boththe resonant enhancements of the FWM and the attenuations of theexcitation pulse absorption and phase matching effects.

$\begin{matrix}{{I_{TFG} = {\frac{2\; \pi^{2}c^{2}\omega_{4}^{2}}{\hslash^{2}\omega_{1}\omega_{2}\omega_{3}^{4}}\frac{n_{1}n_{2}}{n_{3}}\frac{\alpha_{1}\alpha_{2}l^{2}}{\Gamma_{vg}\Gamma_{v^{\prime}g}}\frac{\sigma}{d\; \Omega}}}{\frac{\begin{matrix}{\left( {1 - ^{{- {({\alpha_{1} + \alpha_{2}})}}{l/2}}} \right)^{2} +} \\{4\; ^{{- {({\alpha_{1} + \alpha_{2}})}}{l/2}}\sin {{\Delta \; {{kl}/2}}}^{2}}\end{matrix}}{\left( {\left( {\alpha_{1} + \alpha_{2}} \right){l/2}} \right)^{2} + \left( {\Delta \; {kl}} \right)^{2}}I_{1}I_{2}I_{3}}} & \left( {{Equation}\mspace{14mu} 7} \right)\end{matrix}$

The intensity depends strongly on al at the resonances. It is linearlyproportional to each al at low absorbance and it saturates at largevalues. The phase matching dependence is more complex. The refractiveindex normally decreases from the visible to the mid-IR so the wavevector of the nonlinear polarization |{right arrow over(k)}_(p)|≡(nω/c)=|{right arrow over (k)}₁+{right arrow over (k)}₂+{rightarrow over (k)}₃| is smaller than the {right arrow over (k)}₄ outputfield so |Δ{right arrow over (k≡)}{right arrow over (k)}₄−{right arrowover (k)}_(p)|>0. The anomalous dispersion at the vibrational resonanceshas additional impact on the phase matching and can improve or degradethe phase matching at different points in the resonance. If Δkl is largeenough, small asymmetries should appear in the two-dimensional spectralline shape of the transitions. However, since this may occur at longerpath lengths than those used in these experiments, and since the TSFCMDS line shapes in the experimental spectra are symmetrical and slicesalong ω₁ remained relatively consistent for path lengths of 15-150 μm(see FIG. 8A), the effects of the M-factor in frequency and delay scansimulations are neglected.

Spectral and temporal data were modeled by using the approach describedby Domcke that numerically integrates Equation 1 under the rotating waveapproximation to obtain the evolution of the density matrix elementseventually resulting in the ρ_(eg) output coherence. (See, Gelin, M. F.;Pisliakov, A. V.; Egorova, D.; Domcke, W. A Simple Model for theCalculation of Nonlinear Optical Response Functions and FemtosecondTime-resolved Spectra. J. Chem. Phys. 2003, 118, 5287-5301 and Gelin, M.F.; Egorova, D.; Domcke, W. A New Method for the Calculation ofTwo-pulse Time- and Frequency-resolved Spectra. Chem. Phys. 2005, 312,135-143.) In this method, each pixel of a two-dimensional frequency ordelay scan is calculated independently via propagation of the densitymatrix elements in time. Within the pixel, pulses E_(i)(t) areintroduced with Gaussian envelopes E_(i) ^(o)(t) centered at their delaytime τ_(i). They are responsible for transitions between density matrixelements, with m, n, and p subscripts identifying the dephasing rate,transition moment, and frequency of the quantum states defined in FIG.1A. After integration over the entire excitation period, the ρ_(eg)output signal is Fourier transformed to obtain its frequencydistribution. If a monochromator spectrally resolves the outputfrequencies, the monochromator filter function apodizes the output togive the amplitude of that pixel in the two-dimensional plot.

Parameters of the model include the dephasing times; transition dipoles;pulse frequencies, intensity, and duration; monochromator frequency andresolution; and allowed pathways. This model simulated all of the datapresented in order to qualitatively extract lifetimes, transitionfrequencies, and relative transition dipole moments. Parameters wereshared between all plots with the exception of pulse durations (±0.3ps), which are known to vary across the tuning range, between OPAs, andslightly from day to day.

There are important relationships between TSF-CMDS and DOVE, TRIVE, and2D-IR CMDS. FIG. 1 compares the Wave Mixing Energy Level (WMEL) diagramsfor TSF-CMDS (A), DOVE-CMDS (D-F), and Double Quantum 2D-IR or the TRIVEII/IV α/β pathways (B,C). In all cases v is a fundamental vibration, v′is an overtone or combination band, and e is a virtual electronic statedenoted by a dashed line. The numbers designate the excitationfrequencies (not the time orderings) and the solid and dotted linesdesignate ket and bra transitions, respectively. The double arrow is theoutput coherence. The DOVE pathways differ in the time ordering of theinteractions and the states involved in the transitions. Typically, theoutput signal requires that one IR field provides a double quantumexcitation of a combination band or overtone and the second IR fieldprovides a single quantum excitation of a fundamental, such that thefinal output is a single-quantum Raman transition. Pathway 1(D) is avibrationally enhanced Raman transition and is designated the DOVE-Ramanpathway. Pathways 1(E,F) each have two infrared transitions and aredesignated DOVE-IR pathways. Pathways 1(D) and 1(E) are temporallyoverlapped and create quantum level interference effects. Pathway 1(F)can provide rephasing or line-narrowing if the inhomogeneous broadeningeffects of the v and v′ states are correlated. This pathway can betemporally isolated from the other pathways. DOVE-CMDS also differedfrom other FWM pathways in its ability to observe cross-peaks whenmechanical anharmonicity is absent and only electrical anharmonicitycouples modes. TSF-CMDS also shares this attribute. This attribute canbe exploited to provide structural information.

An interesting characteristic of TSF CMDS is its single fully coherentLiouville pathway. In traditional 2D-IR, the {right arrow over(k)}₁−{right arrow over (k)}₂+{right arrow over (k)}₃ or −{right arrowover (k)}₁+{right arrow over (k)}₂+{right arrow over (k)}₃ pathwayscontain parametric and nonparametric processes that destructivelyinterfere. If mechanical anharmonicity is absent, these processesexactly cancel. Similarly, DOVE pathways (D) and (E) in FIG. 1 have thesame time ordering but opposite signs and can cancel if mechanicalanharmonicity is absent. The pathway in FIG. 1F has a different timeordering that is unique and is not canceled if mechanical anharmonicityis absent. Its intensity would then depend on the electricalanharmonicity. Similarly, the TSFCMDS pathway is unique and is notcanceled if the vibrational modes are mechanically harmonic. Theintensity would also then depend on the electrical anharmonicity and canbe large if |(∂^((n))μ)/(∂Q^((n)))>>0|, as can be true for vibronictransitions. Additionally, TSF-CMDS does not involve populations sopopulation relaxation will not create new peaks. The pathway is fullycoherent and should be capable of isolating the coherent dynamics ofcoherence transfer processes.

The symmetry of the states appearing in the TSF spectrum is constrainedby benzene's inversion symmetry. The modes excited by the firstinteraction have ungerade symmetry and are the same as those in theinfrared absorption spectrum. The symmetry of the overtone orcombination band states excited by the second interaction is defined bythe direct product of the overtone or combination band and must havegerade symmetry for a centrosymmetric molecule. The third interactionexcites a double quantum Raman transition where the initial and finalstates have the gerade symmetry required by the gerade symmetry of thepolarizability. Therefore the cross-peak ω₁ frequencies correspond tothe states observed in the infrared spectrum while the ω₂ frequenciescorrespond to the Raman spectra of overtones, combination bands, andstates coupled to them.

By comparison, DOVE-CMDS has two possible symmetry pathways. DOVE-IRdemands both fundamental and overtone/combination band transitionsexcite modes with ungerade symmetry since each transition involvesabsorption from the ground state. The Raman transition is then anallowed transition between two ungerade states. DOVE-Raman, however,accesses double quantum ungerade states in the first transition butsingle quantum gerade states in the second. The third interactionexcites an allowed transition between two gerade states. The result isthat the combination band and overtone states in TSF-CMDS have exactlythe opposite parity from those in DOVE-CMDS. Additionally, the outputRaman transition occurs as an overtone or combination band transition inTSF-CMDS and a fundamental transition in DOVE-CMDS. Note that theovertone and combination band states in TRIVECMDS and 2D-IR doublequantum pathways involve states of the same parity as TSF-CMDS. Notealso that the electronic states involved in the DOVE-CMDS outputtransition have gerade symmetries while those in the TSF-CMDS outputtransition have ungerade symmetries.

Results

FIG. 2 shows the 1D infrared and Raman vibrational spectra of benzene.These spectra have been fit to a sum of Lorentzian line shapes and theparameters for those fits are given in Table 1, with the τ valuecalculated from F using Lorentzian broadening. Fitting the feature at2948 cm⁻¹ required two Lorentzians and attempts to fit this region tojust the two most prominent peaks resulted in migration of the weakerLorentzian to the 2947 cm⁻¹ location. Therefore there truly are twotransitions here, one of which is quite narrow. (Frequencies of Ramanactive C—H modes are inconsistent in the literature, with many sourcesagreeing with the 3047/3061 cm⁻¹ pair that have been measured here, andmany finding 3057/3074 cm⁻¹ instead.

TABLE 1 Lorentzian Properties of Benzene Transitions Herzberg freq Γ(HWHM) intensity type designation (cm⁻¹) (cm⁻¹) τ (ps) (arb) IR ν₁₃1478.3 3.0 1.8 1 ν₄ + ν₁₁ 1527.9 10 0.53 0.14 ν₁₂ 3048 Raman ν₉ + ν₁₄2926.0 5.6 0.94 0.017 2936.0 2.5 2.1 0.0037 2947.4 3.1 1.7 0.040 2ν₁₃2948.5 1.1 4.8 0.035 ν₁₅ 3047.2 9 0.59 0.42 ν₁ 3061.4 4.5 1.2 1

FIG. 3 shows a 2D TSF frequency spectrum of the intensity dependence onω₁ and ω₂ with time delays τ₂₁=1.2 ps=τ₃₁=1 ps and broad-band filters toisolate the output frequency. With this set of time delays, ω₂ interactsfirst and excites the C═C ring-breathing mode at 1478 cm⁻¹, called v₁₃in the Herzberg notation, that will be used throughout this paper. ω₁comes 1.2 ps later and excites an overtone or coupled state at ˜1478 and1584 cm⁻¹. The visible beam interacts 1 ps later and induces the Ramantransition that creates the output beam and returns the system to theground state. As discussed earlier, the output signal containscontributions from both the driven coherence created during theinteractions with the excitation pulses and the free induction decay ofthe coherence resulting after the excitation pulses. The time delays arelong enough that the free induction decay components will dominate.

The main fundamental transition of FIG. 3 is clearly the bright v₁₃mode. As expected, its overtone state is observed where ω₂ is also ˜1478cm⁻¹. However, the requirement of the second transition havingfundamental ungerade character offers no fundamental vibrational statefor assignment of the cross-peak at ˜1584 cm⁻¹, i.e. this is not acombination band. In DOVE-CMDS, the v₁₆=1600 cm⁻¹ E_(2g) state isobserved due to its different selection rules, but TSF-CMDS isinsensitive to this symmetry. (See, Donaldson, P. M.; Guo, R.; Fournier,F.; Gardner, E. M.; Barter, L. M. C.; Barnett, C. J.; Gould, I. R.;Klug, D. R.; Palmer, D. J.; Willison, K. R. Direct Identification andDecongestion of Fermi Resonances by Control of Pulse Time Ordering inTwo-dimensional IR Spectroscopy. J. Chem. Phys. 2007, 127, 114513.)Therefore the state observed must be that at the sum frequency 3061 cm⁻¹itself, the v₁ A_(1g) C—H stretch, and access to this state arises fromthe Fermi resonance between the v₁₃ mode and the C—H modes.

It is also important to understand how the spectral features depend onthe time delays between the excitation pulses and on contributions fromdifferent phase matching conditions. Since the ω₁ and ω₂ frequencies areclose to each other, the 2{right arrow over (k)}₁+{right arrow over(k)}₃ and 2{right arrow over (k)}₂+{right arrow over (k)}₃ phasematching conditions can also contribute to the spectrum if spatialdiscrimination from the {right arrow over (k)}₁+{right arrow over(k)}₂+{right arrow over (k)}₃ diagonal peak is not adequate. FIG. 4A isa schematic of how the different coherence pathways change for differentexcitation pulse delay times and time orderings. The abscissa is thetime delay between the beams with frequencies ω₃ and ω₁, τ₃₁≡τ₃−τ₁, andthe ordinate is the time delay between the beams with frequencies ω₂ andω₁, τ₂₁≡τ₂−τ₁. The dotted lines define the regions for each of the sixpossible time orderings of the three excitation pulses. The bottom threeWMEL diagrams show the time orderings and resonances when the timedelays are scanned along the directions of the corresponding green,blue, and red arrows in the schematic.

FIG. 4B shows an example of a 2D scan of the τ₂₁ and τ₃₁ time delays atthe diagonal peak where (ω₁, ω₂)=(1475, 1475) cm⁻¹ when there is nospatial discrimination between the three phase matching contributions.As depicted in FIG. 4A, the vertical feature results from the 2{rightarrow over (k)}₁+{right arrow over (k)}₃ phase matching. It has nodependence on the ω₂ pulse time delay. The long diagonal feature resultsfrom the 2{right arrow over (k)}₂+{right arrow over (k)}₃ phase matchingand it has no dependence on the ω₁ time delay. The peak at theintersection between these phase matching features arises from the{right arrow over (k)}₁+{right arrow over (k)}₂+{right arrow over (k)}₃phase matching and it depends on the presence of all three pulses. Thepeak decays away quickly along the upper left diagonal (see the redarrow in FIG. 4A) because the vg coherence dephases as the ω₂ and ω₃beams are delayed from the ω₁ excitation (see right-most WMEL diagram).It also decays away quickly along the positive horizontal axis (see thegreen arrow in FIG. 4A) because the vg coherence dephases as the ω₃ beamis delayed from the ω₁ and ω₂ excitations (see leftmost WMEL diagram).

To determine an accurate value for the v₁₃ mode anharmonicity, a 2Dfrequency scan was collected with the aperture adjusted to pass the{right arrow over (k)}₁+{right arrow over (k)}₂+{right arrow over (k)}₃beam and a small amount of the 2{right arrow over (k)}₂+{right arrowover (k)}₃ beam. This change makes it possible to determine the smallovertone anharmonicity by directly comparing the difference in peakposition on the same axis with the same calibration.

FIG. 5 shows the result of this experiment. The time delays are similarto those in FIG. 3 but with ω₁ and ω₂ permuted such that ω₂ comes secondto excite the Fermi overtone states. The color bar denotes the signalamplitude, i.e., the square root of the output intensity. The spectrumshows the same two peaks as FIG. 3 and a horizontal feature from thesignal that does not depend on ω₁ and corresponds to the outputcoherence wave vector {right arrow over (k)}_(4′)=2{right arrow over(k)}₂+{right arrow over (k)}₃. The {right arrow over (k)}_(4′) signalreaches a maximum along ω₂ at (2ω_(vg)−δ)/2, where δ is the overtoneanharmonicity since the {right arrow over (k)}₂ beam provides both thefundamental and overtone transitions. The {right arrow over (k)}₄ signalreaches a maximum at ω_(vg)−δ, the frequency of the overtone transition.There should therefore be a shift of δ/2 between the {right arrow over(k)}₄ and {right arrow over (k)}_(4′) central frequencies. The FIG. 5inset shows the expected decrease in the ω₂ peak frequency at thediagonal peak due to its shift toward the anharmonic overtone frequency.

However, this diagonal region still contains contribution from the2{right arrow over (k)}₂+{right arrow over (k)}₃ pathway, so it wasnecessary to isolate the {right arrow over (k)}₄ contribution from{right arrow over (k)}_(4′) to accurately determine its peak value. Thisseparation was achieved by finding the average profile of the horizontalfeature at frequencies removed from the diagonal peak and subtractingthat from the diagonal peak region in order to fit the remaining {rightarrow over (k)}₄ feature. Diagonal peak ω₂ slices were then fit toaccurately define the peak ω₂ frequency as a function of ω₁. Thesecentral values varied smoothly and showed that the peak center occurs at1473.7 cm⁻¹. The difference between the {right arrow over (k)}₄ and{right arrow over (k)}_(4′) peak positions (δ/2) is 3.5 cm⁻¹ so δ=7cm⁻¹. This anharmonicity agrees well with the data in Table 1, since thedifference between the 2*v₁₃ frequency (2956.6 cm⁻¹) and the Ramanovertone (2948.5 cm⁻¹) is an 8 cm⁻¹ anharmonicity. These data providefurther confirmation of the contested assignment of this band.

The Wigner plot in FIG. 6 traces out the overtone and cross-peaklifetimes of the TSF states. The beating on the left side of the 2v₁₃peak is attributed to an unresolved, lower energy mode that is too weakto observe directly but becomes observable as beating when themonochromator eliminates most of the contribution of the prominent 2948cm⁻¹ transition.

Discussion

The two peaks observed in the TSF-CMDS spectra are attributed to a Fermiresonance of the E_(1u) overtone (2v₁₃) with the benzene A_(1g) C—Hstretch mode (v₁). The Fermi resonances of benzene's ungerade C—Hstretch modes are well-studied and play an important role in DOVE-CMDS2D spectra. (See, Donaldson, P. M.; Guo, R.; Fournier, F.; Gardner, E.M.; Barter, L. M. C.; Barnett, C. J.; Gould, I. R.; Klug, D. R.; Palmer,D. J.; Willison, K. R. Direct Identification and Decongestion of FermiResonances by Control of Pulse Time Ordering in Two-dimensional IRSpectroscopy. J. Chem. Phys. 2007, 127, 114513.) Calculations show thatFermi resonances create a tetrad of states where the E_(1u)v₁₂ modemixes with the v₁₃+v₁₆, v₂+v₁₃+v₁₈, and v₃+v₁₀+v₁₈ combination bandsthat appear in infrared spectra. Three of these states are allowed inDOVEFWM spectra of the CH mode region and two are observed. (See,Donaldson, P. M.; Guo, R.; Fournier, F.; Gardner, E. M.; Barter, L. M.C.; Barnett, C. J.; Gould, I. R.; Klug, D. R.; Palmer, D. J.; Willison,K. R. Direct Identification and Decongestion of Fermi Resonances byControl of Pulse Time Ordering in Two-dimensional IR Spectroscopy. J.Chem. Phys. 2007, 127, 114513.) The absence of other peaks indicatestheir minimal v₁₃ and v₁₆ character, as coupling to these fundamentalmodes would be required for significant DOVE signal. Consideration ofthis transition strength along with DFT studies therefore led toassignment of Fermi mixing coefficients to members of the ungerade C—Htetrad.

The states appearing in the TSF-CMDS spectra of the current study areattributed to a similar Fermi resonance between the gerade v₁ A_(1g) C—Hstretch mode and the 2v₁₃ overtone. The E_(1u) symmetry of the v₁₃ moderesults in a symmetric direct product for its overtone of A_(1g)+E_(2g),so the first of these allows Fermi mixing with the C—H stretch at 3061cm⁻¹. It is interesting to note that the C—H stretch at 3047 cm⁻¹ (v₁₂)is E_(2g) and therefore could undergo mixing with the other portion ofthe v₁₃ overtone, and yet this cross-peak does not appear in ourspectra. Assuming v₁ and 2v₁₃ are the only two states involved in theA_(1g) mixing, the Fermi resonance is described by

$\begin{matrix}{{{{\Psi_{+} = {{{\cos \left( \frac{\theta}{2} \right)}{v_{1}\rangle}} + {\sin \left( \frac{\theta}{2} \right)}}}}2\; v_{13}}\rangle} & \left( {{Equation}\mspace{14mu} 8} \right) \\{{{{\Psi_{-} = {{{\cos \left( \frac{\theta}{2} \right)}{{2\; v_{13}}\rangle}} + {\sin \left( \frac{\theta}{2} \right)}}}}v_{1}}\rangle} & \left( {{Equation}\mspace{14mu} 9} \right)\end{matrix}$

where Ψ_(±) are the eigenstates resulting from Fermi mixing betweenstates v₁ and 2v₁₃ and θ is the mixing angle resulting from thecoupling, V_(F), by

$\begin{matrix}{\theta = {{atan}\left( \frac{2\; V_{F}}{\hslash \left( {\omega_{v_{1}} - \omega_{2\; v_{13}}} \right)} \right)}} & \left( {{Equation}\mspace{14mu} 10} \right)\end{matrix}$

The transition dipoles for the eigenstates are given by

{right arrow over (μ)}₊ =c ₁{right arrow over (μ)}_(v) ₁ +c ₂{rightarrow over (μ)}_(2v) ₁₃   (Equation 11)

{right arrow over (μ)}⁻ =c ₁{right arrow over (μ)}_(2v) ₁₃ +c ₂{rightarrow over (μ)}_(v) ₁   (Equation 12)

where c₁=cos(θ/2) and c₂=sin(θ/2).

The TSF enhancement in the ρ_(vg)→ρ_(v′g) transition relies dominantlyon the overtone 2v₁₃ absorption strength, while the ρ_(v′g)→ρ_(eg)transition relies dominantly on the v₁ Raman transition strength.Therefore, since the 2948 cm⁻¹ peak has the lower energy and correspondsto Ψ⁻, the peak intensity scales as |c₁|² in the second resonanceenhancement and |c₂|² in the third resonance enhancement. Similarly the3061 cm⁻¹ peak corresponds to Ψ₊ and scales as |c₂|² in the secondresonance enhancement and |c₁|² in the third resonance enhancement.These relationships between the transition moments are part of thesimulation and the resulting intensities agree well with experiment.

There are two peaks in the Raman spectrum that have the same frequenciesobserved in this example. As described above, their difference inintensity and splitting depend on the V_(F) and Δω. Both quantities areunknown as the overtone frequency prior to Fermi mixing is likely notexactly twice the fundamental frequency. If one assumes the relativeRaman intensities of Ψ_(±) depend entirely upon their {right arrow over(μ)}_(v) ₁ character, then one relationship between V_(F) and Δω can bedefined via the mixing angle θ. A second relationship can be defined byusing the frequencies of Ψ_(±). Solving this system of equations leadsto a coupling energy of 19 cm⁻¹. This results in only a 3 cm⁻¹ shift oneach local mode and therefore the other 4-5 cm⁻¹ of shift on the 2948cm⁻¹ mode is a result of mechanical anharmonicity.

Theoretical calculations of cubic anharmonicity in benzene have beendone in the past by finite difference method by using DFT with a B3LYPfunctional and TZ2P basis set, as well as by analytic derivatives with aHartreeFock DZP potential using Moller-Plesset perturbation theory.Anharmonic force constants between these methods agreed within 5% forthe Fermi couplings of the v₁₃ overtone with both the v₁ and v₁₂ C—Hstretches, and resulted in coupling energies of 25 cm⁻¹ for the formerand 12 cm⁻¹ for the latter. This general trend is in agreement with theobservation here of only a v₁ cross-peak. The fact that the v₁₂ state isnot detected at all, may be due to a combination of the weak couplingfrom the v₁₃ fundamental, the possibility of error in that predictedcoupling, the weaker Raman transition strength of v₁₂, and therelatively broad spectral bandwidth of the pulses used.

The Wigner plot and simulation in FIG. 6 trace out the overtone andcross-peak lifetimes of the TSF states. Experimental error in settingthe time delay zero point led to an offset between the experimental andsimulated data. To match the data, the τ₂₁ and τ₃₁ time delays in thesimulated data were translated by 1 ps relative to experimental data.Similar errors are also responsible for changes in relative intensityobserved between the two peaks in FIGS. 6 and 3. This difference arisesbecause the 1584 cm⁻¹ peak intensity relies on the ω₂ interactionoccurring after the ω₁ interaction while the 1478 cm⁻¹ peak hascontributions from both time orderings. The relative intensitiestherefore depend on τ₂₁. In addition, the differences in the dephasingtimes of the states created by the second resonance make the relativeintensities dependent on τ₃₁ so the relative intensities are sensitiveto errors in both delay times. In the Wigner plot, lifetimes predictedby the Raman linewidths reported in Table 1 are in good agreement withexperimental data. The beating observed on the red side of theovertone-dominated Fermi resonance state is attributed to the presenceof a secondary state. It is notable that the 2v₁₃ overtone has asurprisingly long dephasing time of 4.5 ps, as compared to the v₁₃fundamental lifetime of 1.8 ps. It should be noted that assignment ofthis Raman peak was debated in the literature and thought to be acombination band unrelated to the v₁₃ state: v₂+v₃+v₈. (See, Gee, A. R.;Robinson, G. W. Raman Spectrum of Crystalline Benzene. J. Chem. Phys.1967, 46, 4847-4853.) However, the strong coupling between this mode andthe fundamental, as well as the match of unusual lifetime between theRaman and TSF spectra, makes the assignment definitive.

It was surprising that the TSF-CMDS intensity is larger than thecomparable DOVE-CMDS signal, especially since the {right arrow over(k)}₄={right arrow over (k)}₁+{right arrow over (k)}₂+{right arrow over(k)}₃ phase matching condition cannot be phase matched for normaldispersion. To understand this result, simple models incorporating theM-factor using known values for frequency-dependent indices ofrefraction and absorption coefficients were undertaken. The resultsdemonstrate that the benzene peak is not affected by phase mismatch in away that significantly distorts the line shape (FIG. 7) or attenuatesthe signal at long path lengths (FIG. 8).

FIG. 7 shows a set of simulations that model the effects of absorptionand anomalous dispersion on the CMDS efficiency and spectral line-shapesusing Equation 7 and the benzene refractive indices and absorptioncoefficients measured by Bertie. (See, Bertie, J. E.; Jones, R. N.;Keefe, C. D. Infrared Intensities of Liquids 0.12. Accurate OpticalConstants and Molar Absorption Coefficients Between 6225 and 500 cm⁻¹ ofBenzene at 25° C., from Spectra Recorded in Several Laboratories. Appl.Spectrosc. 1993, 47, 891-911 and Bertie, J. E.; Lan, Z. The RefractiveIndex of Colorless Liquids in the Visible and Infrared—Contributionsfrom the Absorption of Infrared and Ultraviolet Radiation and theElectronic Molar Polarizability Below 20500 cm⁻¹ . J. Chem. Phys. 1995,103, 10152-10161.) It assumes fixed field strengths and that ω₁ providesthe fundamental transition and ω₂ the overtone. It is not a numericalintegration of the Schrodinger equation and does not take into accountspectral width of the exciting fields, causing its narrowness relativeto FIG. 3. The absorption coefficients in the numerator of Equation 7create a resonant enhancement that offsets the attenuation described bythe M-factor. The importance of the M-factor depends on the sample pathlength, the absorption coefficients, and the phase mismatch. The modelassumes that insignificant population is removed from the ground state,such that excited state absorption provides the multiplicativeenhancement effects as α₂′, but that the field-depleting absorptioneffects follow the ground state absorption profile even on the ω₂ axis(α₂ in e^(−α) ² ). The absorption from the exponential term, e^(−α) ¹ ,in Equation 7 causes attenuation of the peak intensity while theanomalous dispersion from the v₁₃ mode causes attenuation in the ω₁>1478cm⁻¹ region because of the larger phase-mismatch. The depletion on ω₂occurs at a displaced value from the TSF peak due to anharmonicity, andtherefore does not affect line shape in this instance.

FIG. 7A demonstrates TSF overtone peak line shape based solely upon theabsorption coefficient enhancement that appears in Equation 5 with noM-factor. FIG. 7B is similar but now does take into account the M-factor(at 15 μm, with ±10° phase-matching angles on ω₁ and ω₂) in addition tothe enhancements.

It is also important to understand how the intrinsic phase mismatch andthe absorption effects in Equation 7 together control the dependence ofthe TSF intensity on sample path length. FIG. 8 shows both experimentaland simulated data for the path length dependence. FIG. 8A shows slicescollected along ω₁ with ω₂ at 1480 cm⁻¹ and the pulses temporallyoverlapped. FIG. 8B uses Equation 7, the parameters summarized in Table1, and the published molar absorptivity data to model the intensitydependence of the signal on path length. (See, Bertie, J. E. JohnBertie's Download Site. 2011;http://www.ualberta.ca/jbertie/JBDownload.HTM.) The simulation predictsthat the TSF intensity rises rapidly over a distance that depends on theabsorption at the two excitation frequencies. The intensity becomesconstant at longer path lengths where the excitation length isindependent of the sample thickness and is instead defined by theattenuation of the excitation beams. The attenuation of the excitationbeams shortens the path length over which the phase mismatch develops sothe effects of the phase mismatch are mitigated. The simulation predictsthat the initial intensity increase for the two peaks occurs over a pathlength of ˜10 μm. For the peak at (ω₁, ω₂)=(1584, 1478) cm⁻¹, the ω₂excitation beam is attenuated by the v₁₃ absorption while the ω₁excitation beam is not attenuated. For the peak at (ω₁, ω₂)=(1470, 1478)cm⁻¹, the ω₂ excitation beam is again attenuated by the v₁₃ absorption.The ω₁ excitation beam is resonant with the overtone transition so it isshifted by the anharmonicity. The line width for this transition issufficiently narrow and the anharmonic shift is sufficiently large thatthe simulation predicts the ω₁ excitation beam is not appreciablyattenuated. This prediction is supported by the experimental data. Thepeak intensity at (1470, 1478) cm⁻¹ does not depend on the path length.Although there are changes in the (1584, 1478) cm⁻¹ peak intensity withpath length, the variation is not definitive.

The simulations presented in this example do not consider higher orderwave mixing, preresonance enhancement from electronic states, andelectronic coupling or other Fermi mixing effects.

Example 2 Triply Resonant Sum Frequency Spectroscopy of a Dye Molecule

This example is derived from Boyle et al. Triply Resonant Sum FrequencySpectroscopy: Combining Advantages of Resonance Raman and 2D-IR,submitted to J. Phys. Chem. A and accompanying supporting information,each which is hereby incorporated by reference in its entirety.

Introduction

Triply Resonant Sum Frequency (TRSF) spectroscopy is a fully coherentfour-wave mixing technique that uses the phase matching condition {rightarrow over (k)}_(s)={right arrow over (k)}₁+{right arrow over(k)}₂+{right arrow over (k)}₃ where the subscripts denote the threeexcitation frequencies, ω₁, ω₂, and ω₃. The subscripts do not describethe pulse time ordering since any ordering is possible. Unlike otherfour wave mixing methods where multiple coherence pathways interfere,defining a TRSF time ordering also defines a unique pathway. Forexample,

represents a unique coherence pathway where ω₁ excites fundamentalvibrational modes, ω₂ excites overtone and combination band modes, ω₃excites the final electronic coherence, and the electronic coherencecreates a cooperative and coherent resonance Raman transition returningthe system to the ground state. The fundamental vibrational frequenciesappear on the ω₁ axis and overtones/combination bands appear on the ω₂axis. The anharmonic coupling appears as an offset from the diagonal.

In the steady state, the TRSF output coherence would be given by thedensity matrix element

$\begin{matrix}{\rho_{eg} = \frac{\Omega_{vg}\Omega_{v^{\prime}v}\Omega_{{ev}^{\prime}}}{8\; \Delta_{vg}^{(1)}\Delta_{v^{\prime}g}^{({1,2})}\Delta_{eg}^{({1,2,3})}}} & {{Equation}\mspace{14mu} 13}\end{matrix}$

where Ω_(ba) is the Rabi frequency of the a→b transition, Δ_(ba)^((i,j,k))≡ω_(ba)−ω₁−ω₁−iΓ_(ba), and ω_(ba) and Γ_(ba) are the frequencydifference and dephasing rate of the ba coherence. To gain an intuitiveunderstanding it is possible to take advantage of the directrelationship of the resonances in this equation to absorptioncoefficients and Raman cross-sections. When on-resonance, the outputintensity can be more simply described by the proportionality,

$\begin{matrix}{\left. I_{{T{(R)}}{SF}} \right.\sim\frac{\alpha_{1}\alpha_{2}\sigma_{v^{\prime}g}l^{2}M}{\Gamma_{vg}\Gamma_{v^{\prime}g}}} & {{Equation}\mspace{14mu} 14}\end{matrix}$

where α_(i) is the absorption coefficient of the fundamental orovertone/combination band transitions, σ_(v′g) is the Ramancross-section of the v′g→gg transition, and l is the path length. M is acorrection factor for absorption and phase matching changes.

$\begin{matrix}{M = \frac{{^{{- \alpha_{4}}l}\left( {1 - ^{\frac{\Delta \; \alpha \; l}{2}}} \right)} + {4\; ^{\frac{\Delta \; \alpha \; l}{2}}{\sin^{2}\left( \frac{\Delta \; {kl}}{2} \right)}}}{\left( \frac{\Delta \; \alpha \; l}{2} \right)^{2} + \left( {\Delta \; {kl}} \right)^{2}}} & {{Equation}\mspace{14mu} 15}\end{matrix}$

where α_(i) are the absorption coefficients at the i^(th) excitation oroutput frequency, Δα=α₄−(α₁+α₂+α₃) and Δk is the phase mismatch of thewave vectors. For the TRSF pathway, Δ{right arrow over (k)}≡{right arrowover (k)}₁+{right arrow over (k)}₂+{right arrow over (k)}₃−{right arrowover (k)}₄ where

${k_{i} = \frac{n_{i}\omega_{i}}{c}},$

n_(i) is the index of refraction of the i^(th) frequency, and c is thespeed of light. Equation 15 predicts a decrease in signal as the outputk₄ is absorbed, and saturation of nonlinear gain at the path lengthwhere inputs k₁, k₂, and k₃ have been absorbed.

In the limit of negligible absorption, the M factor reduces to thefamiliar sinc²(kl/2) dependence. The TRSF pathway cannot be phasematched in systems with normal dispersion since the indices ofrefraction make the {right arrow over (k)}₄ output wave vector largerthan the {right arrow over (k)}+{right arrow over (k)}₂+{right arrowover (k)}₃ nonlinear polarization that creates it. Consequently, untilthe present application it hasn't been used for CMDS experiments.Unexpectedly, the inventors have observed high output intensities. Theinventors have determined the high output intensity can be explained atleast in part, because the three resonance enhancements aremultiplicative and create a high nonlinear gain over a path lengthshorter than the inverse of their phase-mismatch, Δk. This gain(Equation 14) relies upon the brightness of coupled infrared modes (α₁,α₂) and the Raman cross-section of the resulting overtone or combinationband (σ). The high gain reduces the effects of absorption and phasemismatch so they do not play a role in these experiments.

As discussed below, TRSF is demonstrated on the donor-acceptor styrylionic dye2-(6-(p-dimethylaminophenyl)-2,4-neopentylene-1,3,5-hexatrienyl)-3-ethylbenzothiazoliumperchlorate), or Styryl 9M.

Experimental.

The CMDS experimental system used a 1 kHz Ti:Sapphire regenerativeamplifier to pump two independently tunable optical parametricamplifiers to create infrared frequencies ω₁ and ω₂. Residual pump lightfrom one OPA provided the ω₃ pulse. The three pulses were focused intothe sample using a linear phase matching geometry where ω₁ and ω₂ weredisplaced ±10° from a central ω₃ pulse, and the ω₄ output was collinearwith ω₃. The focal region had a spot size <100 μm. The ω₁ and ω₂frequencies were independently scanned over the 1250-1700 cm⁻¹ rangewith ω₃=800 nm. Residual ω₃ light was removed by a holographic filterstack as well as a monochromator which resolved the ω₄ frequency with aresolution of ˜100 cm⁻¹. A photomultiplier measured the TRSF outputsignal over the 628-668 nm wavelength range. These frequencies areresonant with the red side of the Styryl 9M electronic transition.

The spectral and temporal pulse widths were ˜15 cm⁻¹ and ˜1.2 ps FWHMfor the ω₁ pulse and ˜18 cm⁻¹ and ˜1.0 ps FWHM for the ω₂ pulse. Thepulse energies were ˜0.5-1 μJ/pulse. The ω₃ pulse was ˜2 ps FWHM andpulse energies between 0.5 and 5 mW. Scanning ω₁ and ω₂ and measuringthe output beam intensity creates two dimensional spectra while scanningthe time delays between the pulses measures the coherent dynamics. Twodimensional Wigner plots of the frequency and time delay show the stateresolved dynamics. The excitation pulse time delays were changed duringspectral scans to correct for the spectral dependence of the pulsetiming. This correction ensured the time delays between pulses remainedconstant.

Styryl 9M was obtained as LDS821 from Exciton. A schematic of the TRSFexcitation scheme is shown in FIG. 15.

Results/Discussion

FTIR absorption spectra of a high concentration Styryl 9M solution withthe solvent contribution removed (3 mM Styryl 9M, 200 μm path length)and the low concentration sample used for the TRSF spectrum (300 μMStyryl 9M with 180 mM benzene in DACN solution, 25 μm path length) wereobtained. The latter sample included benzene at 100 x higherconcentration than Styryl 9M in order to compare molecules with andwithout resonant electronic states. It also had a shorter path length,which together with the lower concentration made the dye absorbance 80times smaller. Since the representative Styryl 9M 1430 cm⁻¹ mode has amolar absorptivity of ˜830 M⁻¹ cm⁻¹, its absorbance would be <0.001 inthe TRSF sample. The benzene line observed at 1480 cm⁻¹ has an opticaldensity of ˜0.04 and a molar absorptivity of 110 M⁻¹ cm⁻¹. The broad1370 cm⁻¹ peak observed is the deuterated acetonitrile solvent (DACN).

A resonance Raman spectrum of Styryl 9M in DACN excited at 514 nm wasalso obtained. The resonance Raman spectrum of the dye contains the samemodes observed in FTIR although the intensities differ. An exception isthat the brightest peak appears at 1522 cm⁻¹ in the FTIR and at 1532cm⁻¹ in the resonance Raman spectrum. Neither feature is Lorentzian andboth fit well to separate modes at 1521 and 1534 cm⁻¹. A UV/Visiblespectrum of 300 μM Styryl 9M with 180 mM benzene in DACN solution, 25 μmpath length was also obtained. The Styryl 9M visible absorption spectrumpeaks at 565 nm, with a FWHM of 132 nm, and ε=7×10⁴ M⁻¹ cm⁻¹.

FIG. 11 shows the TRSF spectrum of 300 μM Styryl 9M, 180 mM benzene inDACN solution and 25 μm path length, obtained with ω₁ and ω₂ temporallyoverlapped and ω₃ delayed 2.5 ps. The ω₃ power used in this scan was ˜2mW. This time ordering should create a symmetric spectrum since eachfield can cause the first or second interaction. Asymmetries in peakwidth and intensity are due to asymmetries in the exciting fields. Allmajor vibrational modes seen in the Raman and FTIR spectra appear in the2D TRSF spectrum, both as diagonal peaks where the interactions involvethe fundamental and overtone transitions and as cross peaks where theinteractions involve the fundamental and combination band transitions.Cross peaks appear between all modes except for one feature at ˜1530cm⁻¹ which notably does not display observable coupling to any othermodes. Monochromator scans show the output frequency appears atω₄=ω₁+ω₂+ω₃ with the ˜25 cm⁻¹ width of the excitation pulses. There isno detectable fluorescence.

A broad background at 1370 cm⁻¹ due to DACN is absent in the TRSFspectrum, and in fact no DACN peaks could be observed in nearby spectralranges when scanning just solvent. The 1480 cm⁻¹ diagonal peak ofbenzene is also absent in FIG. 11 despite having an optical density ˜50times higher. This peak can be observed with higher ω₃ pulse powers insamples without the dye. Similarly, a water impurity has absorbance ˜0.1at 1650 cm⁻¹ that also appears at higher pulse powers or longer pathlengths. These observations support the fact that electronic resonanceenhancement allows detection of vibrational couplings that cannot beseen by infrared absorption or 2D-IR.

The TRSF spectrum of Styryl 9M is quite rich. This dye is extensivelyconjugated and the vibrational modes are delocalized. Previousexperiments show that excitation to the excited electronic state causesreorganization to a Locally Excited (LE) state in <1 ps and to a TwistedIntramolecular Charge Transfer (TICT) state in ˜4 ps. Though TRSF doesnot follow the excited electronic state evolution, the presence of thecross peaks shows that the electronic potential energy surface changesalong the different normal mode coordinates. In Triple Sum Frequency(TSF) experiments without electronic resonances, the overtones andcombination bands appear because of their Fermi resonance withRaman-active fundamental modes. These transitions dominate TSF spectrasince Raman overtone and combination bands are weak.

The spectrum changes when the excitation pulses are not temporallyoverlapped. FIG. 12A-B compares the spectral changes in a section ofFIG. 11 when the time delays are τ₂₁=1.75 ps and τ₃₂=2 ps. Here ω₁ firstexcites the fundamental modes and then ω₂ excites the combinations bandsand overtones. Anharmonic shifts now appear as the frequency differenceof the peaks on the ω₂ axis relative to their frequency on the ω₁ axis.A two-dimensional Gaussian fit of individual peaks in FIG. 12B gives thepeak positions and the resulting anharmonicities (Table 2). Theanharmonicities are smaller than those seen in typical CMDS spectra anddemonstrate the ability of the TRSF pathway to examine peaks withvanishingly small mechanical anharmonicity. The small anharmonicitieswould create very weak 2D-IR features because of the destructiveinterference between coherence pathways.

TABLE 2 Peak frequency maxima of FIG. 12B fit to a 2D gaussian function.This data was also collected reversing the roles of the two OPAs and thedifference was taken as an estimate of experimental error. On the ω₁axis, error ≈1 cm⁻¹; on the ω₂ axis error ≈2 cm⁻¹. ω₁ (cm⁻¹) ω₂ (cm⁻¹)anharmonicity of ω₂ (cm⁻¹) 1531.1 1526.5 −4.6 1511.6 1510.3 −1.3 1477.11509.0 −2.6 1459.1 1513.6 +2.0

Two types of features are left out of Table 2. The first is theprominent antidiagonal mode appearing at 1530 cm⁻¹ in FIG. 12A. Theantidiagonal shape suggests a Δ_(v′g) ^(1,2)=ω_(v′g)−ω₁−ω₂−iΓ_(v′g)factor in Equation 13, where ω_(v′g)˜3060 cm⁻¹. In FIG. 12B this featureresolves into two peaks: (ω₁, ω₂)=(1512,1545) cm⁻¹ and (ω₁,ω₂)=(1531,1527) cm⁻¹. This observation is curious for two reasons. Oneis the lack of cross-mode coupling on the 1530 cm⁻¹ peak, and the otheris the asymmetry of the (1512,1545) cm⁻¹ peak. All other cross peaksappear on both sides of the diagonal because each excitation beam canexcite the fundamental or overtone/combination band transition. Thisasymmetry was also observed in non-electronically resonant TSF ofbenzene. In that study, the Fermi resonance of the C═C ring breathingmode overtone and the C—H stretch fundamental resulted in two mixedmodes. Each mode had overtone character and so each could be accessed inexcited state absorption from the C═C fundamental. For benzene, thisFermi resonance resulted in one “diagonal” peak at (ω₁, ω₂) ˜(1480,1480)cm⁻¹ and one “cross peak” at ˜(1480,1580) cm⁻¹, the sum of which reachthe (primarily) A_(1g) C—H stretch mode at 3060 cm⁻¹. With τ₂₁>1 ps, nopeak appeared at (1580,1480) cm⁻¹ because there was no fundamental modeat 1580 cm⁻¹.

It is therefore proposed that the (1512,1545) cm⁻¹ peak is also causedby a Fermi resonance. This idea requires a fundamental mode at the sumfrequency, 3057 cm⁻¹. The Raman and solvated IR spectra do not provideguidance since it was not possible to detect any modes in the regionsurrounding this frequency of the Raman spectrum due to fluorescence,and the FTIR of the solvated Styryl 9M C—H modes are obscured by solventabsorption. Attenuated total reflectance (ATR) IR shows overlappingpeaks at ˜3064/3072 cm⁻¹ but these can be expected to shift in solvent.The 3057 cm⁻¹ sum frequency is also consistent with that of the diagonalpeak in FIG. 12B at (1531,1527) cm⁻¹, and with the entire antidiagonalfeature in FIG. 12A. The 1512 cm⁻¹ mode is buried in the FTIR and Ramanspectra but becomes significant in TRSF. Since infrared pulses aretemporally overlapped in FIG. 12A, they can excite a two photontransition to the Fermi resonance state where either ω₁ or ω₂ areresonant with the 1512 or 1530 cm⁻¹ fundamentals. A time delay betweenthe infrared pulses defines the time ordering, reduces the number ofpathways, and resolves the antidiagonal feature into the (1531,1527)cm⁻¹ diagonal and the Fermi cross peak at (1512,1545) cm⁻¹. This 1531cm⁻¹ has no observable coupling to other modes in this region. It may bethat this mode has different character than the ring modes representingthe other features in the spectrum.

The second feature is the pair of peaks appearing at (ω₁,ω₂)=(1459,1468) and (1477,1463) cm⁻¹ in FIG. 12B. Only one detectableresonance appears at 1467 cm⁻¹ in this region of FIG. 12A. In the FTIRand resonance Raman spectra, two resonances clearly appeared at 1464.3and 1473.3 cm⁻¹. Since this splitting is within the bandwidth of thelaser pulses used, they would not be resolved in the spectra. The twopeaks in FIG. 12B are attributed to the frequency domain analogue ofquantum beating. It has been previously observed that the quantumbeating between two unresolved states appears as a periodic splittingand collapse of the unresolved peaks. The 9 cm⁻¹ splitting seen in theFTIR and Raman spectra would create frequency domain quantum beatingwith a 3.7 ps period.

In order to measure the dynamics and provide further evidence forspectral quantum beating, a Wigner scan was collected where ω₂ was fixedat 1510 cm⁻¹, τ₃₂ was fixed at 0 ps, and ω₁ was scanned as a function ofτ₁₂ (FIG. 13). When τ₁₂ is positive, the ω₁ pulse arrives after the ω₂and ω₃ pulses. Since the vibronic coherence created by the first twopulses dephases quickly, the coherent output decays rapidly at short τ₁₂times. When τ₁₂ is negative, the dynamics of all the combination bandswith the 1510 cm⁻¹ mode are observed. The quantum beating of the 1464and 1473 cm⁻¹ modes in FIG. 4 is also seen here. When all pulses areoverlapped at τ₁₂=0, these two modes are not resolved, but when ω₁arrives 2 ps prior to ω₂ and ω₃, a gap develops between the two. Atτ₁₂˜−3.3 ps that gap has disappeared again, verifying the splittingfrequency and therefore the origin of the beating behavior. Thissplitting is shown more distinctly in the inset, which was collectedwith the same parameters except τ₃₂ was fixed at 2 ps and greater ω₃power was used. Quantum beating allows mixed-domain spectroscopy toeffectively resolve peaks within the pulse bandwidth in a completelyanalogous manner to time domain spectroscopy.

The detection limit in this example was defined by competition withnonresonant background. The line-shape changes between high and lowconcentrations were examined. The line shape becomes more dispersive asthe nonresonant background begins to dominate at low concentrations. Inthe examination, traces of a diagonal cross-section of the TRSF spectrumin FIG. 11 and traces of the same cross-section when the Styryl 9Mconcentration is 50 μM (i.e. ˜0.15 mOD in the infrared) were compared tosimulations of |χ⁽³⁾|² spectra with a constant nonresonant background,χ_(NR) ⁽³⁾, from the solvent. It was assumed that the frequencydependence can be modeled using a steady state approximation for the twovibrational resonances:

$\begin{matrix}{{\chi^{(3)}\left( {\omega_{1},\omega_{2}} \right)} = {\chi_{NR} + {\sum\limits_{j}^{\;}\; {\sum\limits_{k}^{\;}\; {N\frac{\mu_{j}\mu_{j}\mu_{3}\mu_{4}}{\left( {\delta_{j\; 1} - {\; \Gamma_{j}}} \right)\left( {\delta_{k\; 2} - {\; \Gamma_{k}}} \right)}}}}}} & {{Equation}\mspace{14mu} 16}\end{matrix}$

where j and k are the 17 most important vibrational states,μ_(jk)˜√{square root over (A_(jk)Γ_(jk))}, A is the area of aLorentzian, Γ is its dephasing rate, δ_(jn)=ω_(j)−ω_(n), ω_(n) is thefrequency of the light field, and χ_(NR)=e^(iθ) (represented as constantacross the region and with constant θ assumed between high and lowconcentration). The best match to the data corresponded to θ=35°. Theagreement shows that the line-shape changes are caused by coherentinterference between the resonant and nonresonant contributions to χ⁽³⁾.It has been demonstrated how this model can be used to extract theabsolute values of the real and imaginary parts of χ⁽³⁾ from thedispersive line shapes. The heterodyning from the nonresonant backgroundtherefore represents an extension of the method's detection limits tovery low concentrations.

Conclusions

The electronic resonance in the TRSF pathway provides strongmultidimensional vibrational spectra of the Styryl 9M dye, despiteoptical densities <0.001 in the infrared. The pathway uses two infraredpulses to first excite fundamental and overtone/combination bandvibrational modes, then induces a two-quantum Raman transition using avisible pulse to generate coherent output. Species resonant with thisthird interaction gain significant resonant enhancement above thebackground due to the huge transition dipole moments of electronicstates. This allowed detection of 50 μM Styryl 9M (or ˜0.1 mOD in theinfrared), discrimination against strong solvent and cosoluteabsorption, and multidimensional vibrational spectroscopy of modes withsmall anharmonicity.

The displacement of the excited state potential energy surface alongmultiple normal mode coordinates provides the coupling required forobserving cross-peaks in the 2D spectra. Earlier TSF experiments withoutthe electronic resonance used Fermi resonances with Raman activefundamental modes to create the overtone and combination bandtransitions in the 2D spectra. Not only does the electronic resonanceprovide the coupling required to see multiple modes, it also provides apowerful way to correlate the coupled electronic and vibrational states.Because TRSF has only coherence pathway, coupled modes can be observedwithout the need for mechanical anharmonicity that is required for 2D-IRor TRIVE experiments. Time delays between the first and second infraredinteractions resolves the fundamental and overtone/combination bandsonto orthogonal axes (FIG. 12) so the mechanical anharmonicity appearsas the peak offset from the diagonal in the 2D spectra. This attributeallowed measurement of anharmonicities of order ˜2 cm⁻¹ for Styryl 9M(Table 2), free of the pathway interference that makes suchdeterminations more difficult in 2D-IR and TRIVE. Coherent dynamics andquantum beating can also be observed in TRSF, as demonstrated in theWigner plot in FIG. 13.

Observation of coupling between vibrational and electronic states inthis technique is complementary to other nonlinear spectroscopies suchas full resonant Doubly Vibrationally Enhanced Four-Wave Mixing(DOVE-FWM) and Coherent Anti-Stokes Raman Scattering (CARS). DOVE andT(R)SF experiments are similar because they both probe coupling betweentwo vibrational resonances and one electronic and both are done asmixed-domain spectroscopy. However, DOVE employs the phase-matchingcriterion k_(s)=k₁−k₂+k₃, where k₁ and k₂ are infrared fields and k₃ isvisible or ultraviolet. The k₂ frequency matches a fundamentalvibrational mode and the k₁ an overtone or combination band frequency.Because these vibrations are excited with opposite phase, the visibleexcitation results in a single quantum Raman transition between the twovibrational modes. In the two DOVE-IR pathways, the k₁ and k₂ beams bothcause absorptive transitions from the ground state to vibrational modeswith ungerade character. In the DOVE-Raman pathway, the k₁ beam and thek₂ beams excite a Raman transition where the k₁ beam excites an ungeradeovertone/combination band state and the k₂ beam stimulates emission tothe gerade fundamental vibrational mode. Fully resonant CARS experimentsinvolve resonance with two electronic states and a vibrational state.Both electronic resonances involve ungerade electronic states while thevibrational resonance involves a gerade vibrational state. In contrast,the first interaction in T(R)SF excites states with ungerade fundamentalcharacter and the second interaction excites gerade overtone/combinationband states, similar to 2D-IR or TRIVE.

The electronic enhancement of TRSF spectroscopy is dependent on themolar absorptivity of the electronic transition and the displacement ofthe electronic potential energy surface along the normal modecoordinates. These transitions are much stronger than vibrationaltransitions and greatly enhance the detection limits of CMDS methods.The Styryl 9M dye used in this example has an absorptivity, ε=7×10⁴M⁻¹cm⁻¹, that is typical of electronic resonances in other organicmolecules, metalloproteins, catalysts, etc. This methodology shouldtherefore be a widely applicable technique for obtainingmultidimensional vibrational and electronic spectra at low concentrationwhile also suppressing non-resonant solvent and cosolutes backgroundprocesses.

The word “illustrative” is used herein to mean serving as an example,instance, or illustration. Any aspect or design described herein as“illustrative” is not necessarily to be construed as preferred oradvantageous over other aspects or designs. Further, for the purposes ofthis disclosure and unless otherwise specified, “a” or “an” means “oneor more”.

As will be understood by one skilled in the art, for any and allpurposes, particularly in terms of providing a written description, allranges disclosed herein also encompass any and all possible subrangesand combinations of subranges thereof. Any listed range can be easilyrecognized as sufficiently describing and enabling the same range beingbroken down into at least equal halves, thirds, quarters, fifths,tenths, etc. As a non-limiting example, each range discussed herein canbe readily broken down into a lower third, middle third and upper third,etc. As will also be understood by one skilled in the art, all languagesuch as “up to,” “at least,” “greater than,” “less than,” and the likeincludes the number recited and refers to ranges which can besubsequently broken down into subranges as discussed above. Finally, aswill be understood by one skilled in the art, a range includes eachindividual member.

The foregoing description of illustrative embodiments of the disclosedsubject matter has been presented for purposes of illustration and ofdescription. It is not intended to be exhaustive or to limit thedisclosed subject matter to the precise form disclosed, andmodifications and variations are possible in light of the aboveteachings or may be acquired from practice of the disclosed subjectmatter. The embodiments were chosen and described in order to explainthe principles of the disclosed subject matter and as practicalapplications of the disclosed subject matter to enable one skilled inthe art to utilize the disclosed subject matter in various embodimentsand with various modifications as suited to the particular usecontemplated. It is intended that the scope of the disclosed subjectmatter be defined by the claims appended hereto and their equivalents.

What is claimed is:
 1. A method of obtaining a multidimensional image ofa sample, the method comprising: (a) directing a first coherent lightpulse having a first frequency ω₁ and a first wave vector k₁ at a firstlocation in a sample, (b) directing a second coherent light pulse havinga second frequency ω₂ and a second wave vector k₂ at the first location,(c) directing a third coherent light pulse having a third frequency ω₃and a third wave vector k₃ at the first location and (d) detecting acoherent output signal having a fourth frequency ω₄ and a fourth wavevector k₄ from the first location, wherein ω₄=±ω₁±ω₂±ω₃ andk₄=±k₁±k₂±k₃, wherein at least two of the coherent light pulses each areconfigured to excite a different transition to a discrete quantum stateof a molecule or molecular functionality in the sample, and furtherwherein steps (a)-(d) are repeated at a sufficient number of otherlocations in the sample to provide the multidimensional image.
 2. Themethod of claim 1, wherein ω₄=ω₁+ω₂+ω₃ and k₄=k₁+k₂+k₃.
 3. The method ofclaim 2, wherein the three coherent light pulses are each configured toexcite a different transition to a discrete quantum state in themolecule or molecular functionality.
 4. The method of claim 2, whereinone of the coherent light pulses is configured to excite a transition toa vibrational quantum state, one of the coherent light pulses isconfigured to excite a transition to a different vibrational quantumstate and one of the coherent light pulses is configured to excite atransition to a virtual electronic state, whereby a Raman transition isinduced returning the molecule or molecular functionality to a lowerenergy state.
 5. The method of claim 2, wherein one of the coherentlight pulses is configured to excite a transition to a vibrationalquantum state, one of the coherent light pulses is configured to excitea transition to an electronic quantum state and one of the coherentlight pulses is configured to excite a transition to a virtualelectronic state, whereby a Raman transition is induced returning themolecule or molecular functionality to a lower energy state.
 6. Themethod of claim 3, wherein one of the coherent light pulses isconfigured to excite a transition to a vibrational quantum state, one ofthe coherent light pulses is configured to excite a transition to adifferent vibrational quantum state and one of the coherent light pulsesis configured to excite a transition to an electronic quantum state,whereby a resonance Raman transition is induced returning the moleculeor molecular functionality to a lower energy state.
 7. The method ofclaim 3, wherein one of the coherent light pulses is configured toexcite a transition to a vibrational quantum state, one of the coherentlight pulses is configured to excite a transition to an electronicquantum state and one of the coherent light pulses is configured toexcite a transition to a different electronic quantum state, whereby aresonance Raman transition is induced returning the molecule ormolecular functionality to a lower energy state.
 8. The method of claim2, wherein the frequencies of the at least two coherent light pulses areresonant with their respective transitions.
 9. The method of claim 3,wherein the frequencies of the three coherent light pulses are resonantwith their respective transitions.
 10. The method of claim 2, whereinthe three coherent light pulses interact with the sample to generate thecoherent output signal over a path length and the path length is in therange of from about 1 μm to about 200 μm.
 11. The method of claim 2,wherein the multidimensional image is a three-dimensional image.
 12. Themethod of claim 2, wherein at least two of the coherent light pulses areindependently tunable.
 13. The method of claim 2, wherein the threecoherent light pulses are independently tunable.
 14. The method of claim2, wherein the three coherent light pulses are configured in anon-collinear beam geometry.
 15. A scanning microscope for obtaining amultidimensional image of a sample, the scanning microscope comprising:(a) optics configured to receive coherent light pulses and to direct thecoherent light pulses to a first location in the sample, the coherentlight pulses comprising: (i) a first coherent light pulse having a firstfrequency ω₁ and a first wave vector k₁, (ii) a second coherent lightpulse having a second frequency ω₂ and a second wave vector k₂, and(iii) a third coherent light pulse having a third frequency ω₃ and athird wave vector k₃, wherein at least two of the coherent light pulseseach are configured to excite a different transition to a discretequantum state of a molecule or molecular functionality in the sample;(b) a stage configured to support the sample; and (c) a detectorpositioned to detect a coherent output signal generated from the firstlocation, the coherent output signal having a fourth frequency ω₄ and afourth wave vector k₄, wherein ω₄=±ω₁±ω₂±ω₃ and k₄=±k₁±k₂±k₃, andfurther wherein the scanning microscope is configured to illuminate asufficient number of other locations in the sample with the threecoherent light pulses to provide the multidimensional image.
 16. Thescanning microscope of claim 15, wherein ω₄=ω₁+ω₂+ω₃ and k₄=k₁+k₂+k₃.17. The scanning microscope of claim 16, wherein the optics areconfigured to direct the three coherent light pulses in a non-collinearbeam geometry.
 18. The scanning microscope of claim 16, furthercomprising one or more light sources configured to generate the threecoherent light pulses.
 19. The scanning microscope of claim 18, whereinat least two of the coherent light pulses are independently tunable. 20.The scanning microscope of claim 18, wherein the three coherent lightpulses are independently tunable.